Polymer Density Functional Theory Research Articles

Entropic forces of single-chain confinement in spherical cavities

Thermodynamic properties of a single chain in a confined space have been studied before with the polymer scaling theory and computer simulations. However, a comprehensive understanding of the entropic effects due to the molecular excluded volume and chain connectivity is emerging only recently, especially in the limit of large polymer packing densities as often encountered in biological systems. In this work, we propose a polymer density functional theory (DFT) to study the entropic forces for the confinement of single polymer chains in spherical cavities. At conditions accessible to Monte Carlo simulations, we show that the DFT predictions are in excellent agreement with the simulation results for the distributions of polymer segments as well as the free energy of confinement. The numerical efficiency of the DFT allows us to unify key conclusions from various theoretical analyses and experimental observations.

Density functional theory for predicting polymeric forces against surface fouling

Polymer-based “non-stick” surfaces have been proposed as the next generation of effective and environmentally-friendly coating materials for protecting implanted biomedical devices and for marine antifouling. However, identification of polymeric systems for universal fouling control is often impeded by the poor knowledge of interactions between biological substances and polymeric substrates in diverse aqueous environments. In this article, we review predictions of the polymer density functional theory (DFT) on the structural and surface properties of polymer brushes and polymer nanocomposites that are potentially useful for antifouling applications. In comparison to alternative theoretical methods, DFT exhibits versatile features that are ideal for investigating various polymer-mediated interactions, self-organization of nanoparticles, and surface-induced phase transitions. It is capable of explicit description of important microscopic details of polymeric systems including the molecular excluded-volume effects, associating interactions, van der Waals attraction, Coulomb forces, and inter- and intra- molecular correlations. The theoretical descriptions of surface forces may provide helpful guidelines in the design and development of polymeric materials for preventing non-specific adsorption of biological substances.

Interactions between Surfaces in Polydisperse Semiflexible Polymer Solutions

We generalize a recently developed polymer density functional theory (PDFT) for polydisperse polymer fluids to the case of semiflexible (stiff) polymers. As with our earlier work on flexible polymers, we show here that the generalization of the PDFT to polydisperse mixtures of semiflexible chains allows us to obtain a remarkable simplification of the solution algorithm, compared to the monodispersed case. The number of required equations for complete solution of the (polydisperse) PDFT scales inveresly with the width of the polydispersity profile. This reflects the fact that more information is required, the narrower is the polydispersity. We apply the theory to a number of model scenarios, including repulsive and adsorbing surfaces. We find that polydispersity effects on the depletion interaction are larger for dilute solutions and when the surfaces are adsorbing. We also investigate short, stiff chains and find rather dramatic effects in the presence of adsorbing surfaces.

Colloidal interactions in thermal and athermal polymer solutions: The Derjaguin approximation, and exact results for mono- and polydisperse ideal chains

Polymer density functional theory is used to test the accuracy of the Derjaguin approximation in describing the interaction between two spherical colloidal particles immersed in a polymer solution. The density functional equations are solved numerically in a cylindrical geometry and thus a rigorous comparison is afforded, within the density functional framework. The density functional theory provides an exact (albeit numerical) treatment for ideal polymers. We investigate both nonadsorbing and telechelic chains, as well as consider the behavior of widely polydisperse samples (living chains). We make comparisons with the commonly used penetrable sphere models for polymers and are able to identify strengths and weaknesses in the latter. We find that the Derjaguin approximation performs surprisingly well, even for particle sizes that are comparable to the radius of gyration of the polymers.

Density functional approach for modeling CO2 pressurized polymer thin films in equilibrium

We have used polymer density functional theory to analyze the equilibrium density profiles and interfacial properties of thin films of polymer in the presence of CO(2). Surface tension, surface excess adsorption of CO(2) on polymer surface, and width of the interface are discussed. We have shown the changes in these properties in the presence of CO(2) and with increasing film thickness and their inverse linear relationship with increasing chain length. One of our important findings is the evidence of segregation of end segments toward the interface. We have introduced a new method of representing this phenomenon by means of Delta profiles that show increase in segregation owing to the presence of CO(2) and with increasing chain length. We also make predictions for the octacosane-CO(2) binary system near the critical point of CO(2). Our results indicate qualitative trends that are comparable to the similar experimental and simulation studies.

Simulating Equilibrium Surface Forces in Polymer Solutions Using a Canonical Grid Method

A new simulation method for nonuniform polymer solutions between planar surfaces at full chemical equilibrium is described. The technique uses a grid of points in a two-dimensional thermodynamic space, labeled by surface area and surface separations. Free energy differences between these points are determined via Bennett's optimized rates method in the canonical ensemble. Subsequently, loci of constant chemical potential are determined within the grid via simple numerical interpolation. In this way, a series of free energy versus separation curves are determined for a number of different chemical potentials. The method is applied to the case of hard sphere polymers between attractive surfaces, and its veracity is confirmed via comparisons with established alternative simulation techniques, namely, the grand canonical ensemble and isotension ensemble methods. The former method is shown to fail when the degree of polymerization is too large. An interesting interplay between repulsive steric interactions and attractive bridging forces occurs as the surface attraction and bulk monomer density are varied. This behavior is further explored using polymer density functional theory, which is shown to be in good agreement with the simulations. Our results are also discussed in light of recent self-consistent field calculations which correct the original deGennes results for infinitely long polymers. In particular, we look at the role of chain ends by investigating the behavior of ring polymers.

A theoretical study for nanoparticle partitioning in the lamellae of diblock copolymers

Morphology control is important for practical applications of composite materials that consist of functional polymers and nanoparticles. Toward that end, block copolymers provide useful templates to arrange nanoparticles in the scaffold of self-organized polymer microdomains. This paper reports theoretical predictions for the distribution of nanoparticles in the lamellar structures of symmetric diblock copolymers on the basis of a polymer density functional theory (DFT) and the potential distribution theorem (PDT). The DFT predicts periodic spacing of lamellar structures in good agreement with molecular dynamics simulations. With the polymer structure from DFT as the input, the PDT is used to examine the effects of particle size, surface energy, polymer chain length, and compressibility on the distribution of nanoparticles in the limit of low particle density. It is found that the nanoparticle distribution depends not only on the particle size and surface energy but also on the local structure of the microdomain interface, polymer chain length, and compressibility. The theoretical predictions are compared well with experiments and simulations.

Simulations and density functional calculations of surface forces in the presence of semiflexible polymers

We simulate interactions between adsorbing and nonadsorbing surfaces immersed in solutions containing monodisperse semiflexible chains. Apart from the nature of the surfaces, we investigate responses to changes of the intrinsic chain stiffness, the degree of polymerization, and the bulk concentration. Our simulations display a sufficient accuracy and precision to reveal free-energy barriers that are small on a typical scale of surface force simulations, but still of the same order as the expected van der Waals interactions. Two different approaches have been tested: grand canonical simulations, improved by configurational-biased techniques, and a perturbation method utilizing the isotension ensemble. We find the former to be preferable when the surfaces are nonadsorbing, whereas the isotension approach is superior for calculations of interactions between adsorbing surfaces, especially if the polymers are stiff. We also compare our simulation results with predictions from several versions of polymer density functional theory. We find that a crucial aspect of these theories, in quantitative terms, is that they recognize that end monomers exclude more volume to the surrounding than inner ones do. Those theories provide satisfactorily accurate predictions, particularly when the surfaces are nonadsorbing.

Density functional theory for flexible and semiflexible polymers of infinite length

We develop a density functional theory for infinitely long flexible and semiflexible polymers, as an appropriate simplification of the corresponding treatment of finite chains. We demonstrate that very long chains sometimes are required to reproduce even qualitatively correct limiting behaviors.

Surface Forces between Telechelic Brushes Revisited: The Origin of a Weak Attraction

Telechelic polymers are useful for surface protection and stabilization of colloidal dispersions by the formation of polymer brushes. A number of theoretical investigations have been reported on a weak attraction between two telechelic brushes when they are at the classical contact, i.e., when the surface separation is approximately equal to the summation of the brush thicknesses. While recent experiments have confirmed the weak attraction between telechelic brushes, its origin remains elusive because of conflicting approximations used in the previous theoretical calculations. In this paper, we have investigated the telechelic polymer-mediated surface forces by using a polymer density functional theory (PDFT) that accounts for both the surface-adhesive energy and segment-level interactions specifically. Within a single theoretical framework, the PDFT is able to capture both the depletion-induced attraction in the presence of weakly adhesive polymers and the steric repulsion between compressed polymer brushes. In comparison of the solvation forces between telechelic brushes with those between brushes formed by surfactant-like polymers and with those between two asymmetric surfaces mediated by telechelic polymers, we conclude that the weak attraction between telechelic brushes is primarily caused by the bridging effect. Although both the surfactant-like and telechelic polymers exhibit a similar scaling behavior for the brush thickness, a significant difference has been observed in terms of the brush microstructures, in particular, the segment densities near the edges of the polymer brushes.

Polymer density functional theory approach based on scaling second-order direct correlation function

A second-order direct correlation function (DCF) from solving the polymer-RISM integral equation is scaled up or down by an equation of state for bulk polymer, the resultant scaling second-order DCF is in better agreement with corresponding simulation results than the un-scaling second-order DCF. When the scaling second-order DCF is imported into a recently proposed LTDFA-based polymer DFT approach, an originally associated adjustable but mathematically meaningless parameter now becomes mathematically meaningful, i.e., the numerical value lies now between 0 and 1. When the adjustable parameter-free version of the LTDFA is used instead of the LTDFA, i.e., the adjustable parameter is fixed at 0.5, the resultant parameter-free version of the scaling LTDFA-based polymer DFT is also in good agreement with the corresponding simulation data for density profiles. The parameter-free version of the scaling LTDFA-based polymer DFT is employed to investigate the density profiles of a freely jointed tangent hard sphere chain near a variable sized central hard sphere, again the predictions reproduce accurately the simulational results. Importance of the present adjustable parameter-free version lies in its combination with a recently proposed universal theoretical way, in the resultant formalism, the contact theorem is still met by the adjustable parameter associated with the theoretical way.

Polymer density functional approach to efficient evaluation of path integrals

A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures. The exact solution is not, though, reachable in three dimensions (3D) because of a vast amount of storage required for 2p-PCF. In order to treat closed paths in 3D, we introduce a so-called "open ring" approximation which proves to be rather accurate in the limit of long chains. We also employ a simple self-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen and helium, and ions of He and Li. Our results compare rather well to the known data, while the computational effort (some seconds or minutes) is about 100 times less than that with Monte Carlo simulations. Moreover, the well-known "sign problem" is expected to be considerably reduced within the reported PI-DFT, since it allows for a direct estimate of the corresponding partition functions.

Density functional approach for inhomogeneous star polymer fluids

We propose microscopic density functional theory for inhomogeneous star polymer fluids. Our approach is based on fundamental measure theory for hard spheres, and on Wertheim's first- and second-order perturbation theory for the interparticle connectivity. For simplicity we consider a model in which all the arms are of the same length, but our approach can be easily extended to the case of stars with arms of arbitrary lengths.

Surface-induced phase transitions in ultrathin films of block copolymers

We study theoretically the lamellar-disorder-lamellar phase transitions of AB diblock and tetrablock copolymers confined in symmetric slitlike pores where the planar surface discriminatingly adsorbs A segments but repels B segments, mimicking the hydrophobic/hydrophilic effects that have been recently utilized for the fabrication of environmentally responsive "smart" materials. The effects of film thickness, polymer volume fraction, and backbone structure on the surface morphology have been investigated using a polymer density-functional theory. The surface-induced phase transition is manifested itself in a discontinuous switch of microdomains or a jump in the surface density dictated by the competition of surface adsorption and self-aggregation of the block copolymers. The surface-induced first-order phase transition is starkly different from the thickness-induced symmetric-asymmetric or horizontal-vertical transitions in thin films of copolymer melts reported earlier.

Surface forces in polymer fluids: a comparison between simulations and density functional theory.

A polymer density functional theory is evaluated in terms of its ability to predict interactions between large surfaces in a polymer fluid. Comparisons are made with results from simulations in an expanded isotension ensemble. The variation of the net surface-surface interaction with adsorption strength is examined. Cases where the monomers interact via a pure hard sphere potential are investigated, but we have also studied the effect of attractions between the monomers. In all cases, we obtain an almost quantitative agreement between the simulated results and the predictions from the polymer density functional theory.

An improved density functional description of hard sphere polymer fluids at low density

A refined formulation of an existing polymer density functional theory is presented, wherein an intrachain stiffness is introduced via a bending potential. Comparisons with Metropolis Monte Carlo simulations in a slit geometry shows that this leads to a considerable improvement of the predicted density profile for a hard sphere polymer melt, at low density. We also show how the corresponding surface interactions are affected by the inclusion of this intramolecular correlation. We expect that the improvement obtained will be even more important in the description of, for example, polyelectrolytes, although such comparisons are not made in this preliminary study.

Density functional theory for nonuniform polymers: Accurate treatment of the effect of attractive interactions

A density functional theory is presented for the effect of fluid–fluid and fluid–surface attractive interactions on the structure of polymers at surfaces. The theory treats the ideal gas free-energy functional exactly and uses a weighted density approximation for the hard chain contribution to the excess free-energy functional. The attractive interactions are calculated using the bulk fluid direct correlation function obtained from the polymer reference interaction site model theory. The predictions of the theory are in good agreement with computer simulation results for the density profiles of freely rotating fused-sphere chains at surfaces for a wide range of densities and temperatures. The results emphasize the importance of using different approximations for the hard sphere and attractive interactions in density functional theories for polymers.

A self-consistent-field integral equation theory for nonuniform polymer fluids

An integral equation theory is developed for nonuniform polymer fluids. It is based on an exact formulation of the density functional theory for polymers, which is essentially identical in form to the popular self-consistent-field approximations. A nonuniform site–site Ornstein–Zernike equation is obtained. As well, the Triezenberg–Zwanzig–Wertheim–Lovett– Mou–Buff equations are generalized to polymer fluids. The Percus–Yevick and mean spherical approximations are suggested as plausible closures to these equations.

A density functional theory for polymers: Application to hard chain–hard sphere mixtures in slitlike pores

A density functional theory for polymer fluids is derived. The form of the theory is identical to a continuum self-consistent-field theory but we show that with the correct ‘‘mean field,’’ the theory is exact. We apply the theory to a polymer fluid made up of tangential hard spheres and borrow from a successful density functional theory for simple hard spheres to obtain a nonlocal functional expression for this mean field. The latter depends only upon the monomer density. This approximation is used to treat the problem of a mixture of such polymers and simple hard spheres, contained in slitlike pores. Comparison with simulations show that the theory, though simple in structure, and containing a single adjustable parameter, is surprisingly accurate.