Density functional theory study on the structure and capillary phase transition of a polymer melt in a slitlike pore: effect of attraction

A density functional theory is proposed to investigate the effects of polymer monomer-monomer and monomer-wall attractions on the density profile, chain configuration, and equilibrium capillary phase transition of a freely jointed multi-Yukawa fluid confined in a slitlike pore. The excess Helmholtz energy functional is constructed by using the modified fundamental measure theory, Wertheim's first-order thermodynamic perturbation theory, and Rosenfeld's perturbative method, in which the bulk radial distribution function and direct correlation function of hard-core multi-Yukawa monomers are obtained from the first-order mean spherical approximation. Comparisons of density profiles and bond orientation correlation functions of inhomogeneous chain fluids predicted from the present theory with the simulation data show that the present theory is very accurate, superior to the previous theory. The present theory predicts that the polymer monomer-monomer attraction lowers the strength of oscillations for density profiles and bond orientation correlation functions and makes the excess adsorption more negative. It is interesting to find that the equilibrium capillary phase transition of the polymeric fluid in the hard slitlike pore occurs at a higher chemical potential than in bulk condition, but as the attraction of the pore wall is increased sufficiently, the chemical potential for equilibrium capillary phase transition becomes lower than that for bulk vapor-liquid equilibrium.     

Monte Carlo simulation of the self-assembly and phase behavior of semiflexible equilibrium polymers

Grand canonical Monte Carlo simulations of a simple model semiflexible equilibrium polymer system, consisting of hard sphere monomers reversibly self-assembling into chains of arbitrary length, have been performed using a novel sampling method to add or remove multiple monomers during a single MC move. Systems with two different persistence lengths and a range of bond association constants have been studied. We find first-order lyotropic phase transitions between isotropic and nematic phases near the concentrations predicted by a statistical thermodynamic theory, but with significantly narrower coexistence regions. A possible contribution to the discrepancy between theory and simulation is that the length distribution of chains in the nematic phase is bi-exponential, differing from the simple exponential distribution found in the isotropic phase and predicted from a mean-field treatment of the nematic. The additional short length-scale characterizing the distribution appears to arise from the lower orientational order of short chains. The dependence of this length-scale on chemical potential, bond association constant, and total monomer concentration has been examined.     

Computer simulation study of the global phase behavior of linear rigid Lennard-Jones chain molecules: comparison with flexible models

The global phase behavior (i.e., vapor-liquid and fluid-solid equilibria) of rigid linear Lennard-Jones (LJ) chain molecules is studied. The phase diagrams for three-center and five-center rigid model molecules are obtained by computer simulation. The segment-segment bond lengths are L = sigma, so that models of tangent monomers are considered in this study. The vapor-liquid equilibrium conditions are obtained using the Gibbs ensemble Monte Carlo method and by performing isobaric-isothermal NPT calculations at zero pressure. The phase envelopes and critical conditions are compared with those of flexible LJ molecules of tangent segments. An increase in the critical temperature of linear rigid chains with respect to their flexible counterparts is observed. In the limit of infinitely long chains the critical temperature of linear rigid LJ chains of tangent segments seems to be higher than that of flexible LJ chains. The solid-fluid equilibrium is obtained by Gibbs-Duhem integration, and by performing NPT simulations at zero pressure. A stabilization of the solid phase, an increase in the triple-point temperature, and a widening of the transition region are observed for linear rigid chains when compared to flexible chains with the same number of segments. The triple-point temperature of linear rigid LJ chains increases dramatically with chain length. The results of this work suggest that the fluid-vapor transition could be metastable with respect to the fluid-solid transition for chains with more than six LJ monomer units.     

Theoretical analysis of polydispersity in the nematic phase of self-assembled semiflexible chains

In recent simulations of semiflexible equilibrium polymers [X. Lu and J. T. Kindt, J. Chem. Phys. 120, 10328 (2004)] a roughly biexponential distribution of chain lengths was observed in the nematic phase. In this study, we show that a theory representing the nematic phase as an equilibrium mixture of randomly oriented chains below a critical length Lc and chains above Lc obeying a Gaussian orientational distribution reproduces this distribution qualitatively. The agreement between predicted and simulated phase boundaries is improved, with a narrower isotropic-nematic coexistence region, compared with a monoexponential chain distribution. We find, furthermore, that the critical length Lc scales as the inverse of monomer concentration, irrespective of the bond strength of the chains, but that at the phase boundary, the fraction of monomers found in disordered chains peaks at a certain bond strength.     

Phase behavior and structure formation in linear multiblock copolymer solutions by Monte Carlo simulation

The solution phase behavior of short, strictly alternating multiblock copolymers of type (A(n)B(n))(m) was studied using lattice Monte Carlo simulations. The polymer molecules were modeled as flexible chains in a monomeric solvent selective for block type A. The degree of block polymerization n and the number of diblock units per chain m were treated as variables. We show that within the regime of parameters accessible to our study, the thermodynamic phase transition type is dependent on the ratio of m / n. The simulations show microscopic phase separation into roughly spherical aggregates for m / n ratios less than a critical value and first-order macroscopic precipitation otherwise. In general, increasing m at fixed n, or n at fixed m, promotes the tendency toward macroscopic phase precipitation. The enthalpic driving force of phase change is found to universally scale with chain length for all multiblock systems considered and is independent of the existence of a true phase transition. For aggregate forming systems at low amphiphile concentrations, multiblock chains are shown to self-assemble into intramolecular, multichain clusters. Predictions for microstructural dimensions, including critical micelle concentration, equilibrium size, shape, aggregation parameters, and density distributions, are provided. At increasing amphiphile density, interaggregate bridging is shown to result in the formation of networked structures, leading to an eventual solution-gel transition. The gel is swollen and consists of highly interconnected aggregates of approximately spherical morphology. Qualitative agreement is found between experimentally observed physical property changes and phase transitions predicted by simulations. Thus, a potential application of the simulations is the design of multiblock copolymer systems which can be optimized with regard to solution phase behavior and ultimately physical and mechanical properties.     

Surface-induced phase behavior of polymer/nanoparticle blends with attractions

In an athermal blend of nanoparticles and homopolymer near a hard wall, there is a first order phase transition in which the nanoparticles segregate to the wall and form a densely packed monolayer above a certain nanoparticle density. Previous investigations of this phase transition employed a fluids density functional theory (DFT) at constant packing fraction. Here we report further DFT calculations to probe the robustness of this phase transition. We find that the phase transition also occurs in athermal systems at constant pressure, the more natural experimental condition than constant packing fraction. Adding nanoparticle-polymer attractions increases the nanoparticle transition density, while sufficiently strong attractions suppress the first-order transition entirely. In this case the systems display a continuous transition to a bulk layered state. Adding attractions between the polymers and the wall has a similar effect of delaying and then suppressing the first-order nanoparticle segregation transition, but does not lead to any continuous phase transitions.     

Chain conformations and phase separation in polymer solutions with varying solvent quality

Molecular dynamics simulations are used to investigate the conformations of a single polymer chain, represented by the Kremer-Grest bead-spring model, in a solution with a Lennard-Jones liquid as the solvent when the interaction strength between the polymer and solvent is varied. Results show that when the polymer-solvent interaction is unfavorable, the chain collapses as one would expect in a poor solvent. For more attractive polymer-solvent interactions, the solvent quality improves and the chain is increasingly solvated and exhibits ideal and then swollen conformations. However, as the polymer-solvent interaction strength is increased further to be more than about twice the strength of the polymer-polymer and solvent-solvent interactions, the chain exhibits an unexpected collapsing behavior. Correspondingly, for strong polymer-solvent attractions, phase separation is observed in the solutions of multiple chains. These results indicate that the solvent becomes effectively poor again at very attractive polymer-solvent interactions. Nonetheless, the mechanism of chain collapsing and phase separation in this limit differs from the case with a poor solvent rendered by unfavorable polymer-solvent interactions. In the latter, the solvent is excluded from the domain of the collapsed chains while in the former, the solvent is still present in the pervaded volume of a collapsed chain or in the polymer-rich domain that phase separates from the pure solvent. In the limit of strong polymer-solvent attractions, the solvent behaves as a glue to stick monomers together, causing a single chain to collapse and multiple chains to aggregate and phase separate.     

Precipitation of oppositely charged polyelectrolytes in salt solutions

We study phase separation in symmetric solutions of weakly charged flexible chains of opposite sign. Precipitation is caused by effective attractions due to charge fluctuations and by short-range attractions between monomers. The contribution from charge fluctuations is computed within the random phase approximation (RPA), which takes into account the connectivity of charges in the polyions. The impenetrability of the ions is accounted for by using a modified Coulomb potential in the RPA. In good solvent conditions the precipitate monotonically swells and eventually dissolves upon addition of salt. However, near the theta-solvent condition, but still in the good solvent, the precipitate can be stable at any salt concentration. Moreover, the density of the precipitate after initial decrease can increase with addition of salt. This effect is a result of redistribution of salt between the precipitate and the supernatant, which is due to an interplay of electrostatic and hardcore interactions. For not too weakly charged polyions the precipitate properties become strongly dependent on temperature even in good solvent conditions.     

Onsager chains: Semi-flexible polymers revisited

Using a density functional approach we derive the equations describing the equilibrium orientational distribution of a system of chains composed of elongated segments that interact with segments located on other chains through excluded volume interactions and with neighbouring segments of the same chain through a potential that determines the chain flexibility. We analytically determine the limit of stability of the low density isotropic phase as a function of the number of segments and the chain stiffness. The approach turns out to be formally equivalent to a recently proposed mean-field theory by Petschek and Terentjev. Comparison with the Khoklov-Semenov theory shows that the latter is based on an additional assumption that is not valid in an orientationally ordered phase.     

Molecular simulation study of cavity-generated instabilities in the superheated Lennard-Jones liquid

Previous equilibrium-based density-functional theory (DFT) analyses of cavity formation in the pure component superheated Lennard-Jones (LJ) liquid [S. Punnathanam and D. S. Corti, J. Chem. Phys. 119, 10224 (2003); M. J. Uline and D. S. Corti, Phys. Rev. Lett. 99, 076102 (2007)] revealed that a thermodynamic limit of stability appears in which no liquidlike density profile can develop for cavity radii greater than some critical size (being a function of temperature and bulk density). The existence of these stability limits was also verified using isothermal-isobaric Monte Carlo (MC) simulations. To test the possible relevance of these limits of stability to a dynamically evolving system, one that may be important for homogeneous bubble nucleation, we perform isothermal-isobaric molecular dynamics (MD) simulations in which cavities of different sizes are placed within the superheated LJ liquid. When the impermeable boundary utilized to generate a cavity is removed, the MD simulations show that the cavity collapses and the overall density of the system remains liquidlike, i.e., the system is stable, when the initial cavity radius is below some certain value. On the other hand, when the initial radius is large enough, the cavity expands and the overall density of the system rapidly decreases toward vaporlike densities, i.e., the system is unstable. Unlike the DFT predictions, however, the transition between stability and instability is not infinitely sharp. The fraction of initial configurations that generate an instability (or a phase separation) increases from zero to unity as the initial cavity radius increases over a relatively narrow range of values, which spans the predicted stability limit obtained from equilibrium MC simulations. The simulation results presented here provide initial evidence that the equilibrium-based stability limits predicted in the previous DFT and MC simulation studies may play some role, yet to be fully determined, in the homogeneous nucleation and growth of embryos within metastable fluids.     

Monte Carlo simulations of equilibrium solubilities and structure of water in n-alkanes and polyethylene

Gibbs ensemble Monte Carlo methods based on a force field that combines the simple point charge [Berendsen et al., in Intermolecular Forces, edited by Pullman (Reidel, Dordrecht, 1981), p. 331] and transferable potentials for phase equilibria [Martin and Siepmann, J. Phys. Chem. B 102, 2659 (1998)] models were used to study the equilibrium properties of binary systems consisting of water and n-alkanes with chain lengths from hexane to hexadecane. In addition, systems where extended linear alkane chains (up to 300 carbon units long) were used to represent amorphous polyethylene were simulated in the presence of water using a connectivity altering osmotic Gibbs ensemble. In these simulations the equilibrium between a liquid water phase and a polymer phase into which water was inserted was studied. The predicted solubilities, which were determined between 350 and 550 K, are in good agreement with experiment, where experimental results are available, and the density of water molecules in the hydrocarbons is approximately 63% as high as in saturated water vapor under the same conditions. At the lower temperatures most of the water exists as monomers; increasing the temperature leads to an increase in the density of water in the alkane phase and hence in the fraction of molecules that participate in clusters. Dimers are the most prevalent clusters in all hydrocarbons and at all temperatures studied, and the fraction of clusters of given size decrease with increasing cluster size. A large fraction of trimers, tetramers, and pentamers, which are the cluster sizes for which topologies have been studied, are cyclic at low temperatures, but at higher temperatures linear structures predominate. The same properties are observed for pure water vapor clusters in equilibrium with the liquid phase, showing that the cluster topologies are not significantly affected by the surrounding hydrocarbon.     

Simulation and theory of self-assembly and network formation in reversibly cross-linked equilibrium polymers

A simulation model of hard spheres capable of reversible assembly into chains, which then may reversibly cross-link into networks, has been studied through grand canonical Monte Carlo simulation. Effects of varying intra- and interchain bond strengths, chain flexibilities, and restrictions on cross-linking angle were investigated. Observations including chain-length distributions and phase separation could be captured in most cases using a simple model theory. The coupling of chain growth to cross-linking was shown to be highly sensitive to the treatment of cross-linking by chain ends. In some systems, ladderlike domains of several cross-links joining two chains were common, resulting from cooperativity in the cross-linking. Extended to account for this phenomenon, the model theory predicts that such cooperativity will suppress phase separation in weakly polymerizing chains and at high cross-link concentration. In the present model, cross-linking stabilizes the isotropic phase with respect to the nematic phase, causing a shift in the isotropic-nematic transition to higher monomer concentration than in simple equilibrium polymers.     

Phase equilibria and plate-fluid interfacial tensions for associating hard sphere fluids confined in slit pores

The excess Helmholtz free energy functional for associating hard sphere fluid is formulated by using a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)]. Within the framework of density functional theory, the thermodynamic properties including phase equilibria for both molecules and monomers, equilibrium plate-fluid interfacial tensions and isotherms of excess adsorption, average molecule density, average monomer density, and plate-fluid interfacial tension for four-site associating hard sphere fluids confined in slit pores are investigated. The phase equilibria inside the hard slit pores and attractive slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal and the plate-fluid interfacial tensions at equilibrium states are predicted consequently. The influences of association energy, fluid-solid interaction, and pore width on phase equilibria and equilibrium plate-fluid interfacial tensions are discussed.     

Density functional theory of homopolymer mixtures confined in a slit

A density functional theory (DFT) is developed for polymer mixtures with shorted-ranged attractive interparticle interactions confined in a slit. Different weighting functions are used separately for the repulsive part and the attractive part of the excess free energy functional by applying the weighted density approximation. The predicted results by DFT are in good agreement with the corresponding simulation data indicating the reliability of the theory. Furthermore, the center-of-mass profiles and the end-to-end distance distributions are obtained by the single chain simulation; the predictions also agree well with simulation data. The results reveal that both the attraction of the slit wall and the temperature has stronger effect on longer chains than on shorter ones because the intrasegment correlation of chains increases with increasing chain length.     

End-evaporation dynamics revisited

We present analytical results on the so-called end-evaporation kinetics in equilibrium polymeric systems following a temperature jump (T jump). A T jump prepares the system with a nonequilibrium length distribution, after which it relaxes back to its equilibrium state. Starting from a master equation, we develop a mean-field analytical theory based on a generating function approach, which allows explicit approximate expressions for the monomer and dimer concentrations to be derived in a discrete setting; the concentrations of the other chains as well as the average chain length were shown to be entirely expressible in terms of the monomer and dimer concentrations. We find that the calculated monomer and dimer concentrations as well as the average chain length are in good agreement with numerical simulation results and do not suffer from some of the defects of earlier continuum theories. Furthermore, the relaxation was shown to take place in three different stages. The first stage comprises the very fast relaxation of the monomers to almost their equilibrium concentration; the other polymer chains have hardly relaxed. During the second stage, which is highly nonlinear, a redistribution of material at practically constant monomer density takes place. Only in the final stage of the relaxation process the chain concentrations approach their true equilibrium values. In this stage there are only very small shifts in the concentrations of chains, which are governed by extremely slow "indirect" monomer-mediated processes.     

Linking phase behavior and reversible colloidal aggregation at low concentrations: simulations and stochastic mean field theory

We have studied the link between the kinetics of clustering and the phase behavior of dilute colloids with short range attractions of moderate strength. This was done by means of computer simulations and a theoretical kinetic model originally developed to deal with reversible colloidal aggregation. Three different regions of the phase diagram were accessed. For weak attractions, a gas phase of small clusters in equilibrium forms in the system. For intermediate attractions, the system undergoes liquid-gas separation, which is signatured by the formation of a few large droplike aggregates, a gas phase of small clusters, and an overall kinetics where a few seeds succeed in explosively growing at long times, after a lag time. Finally, for very strong attractions, fractal unbreakable clusters form and grow following DLCA-like (diffusion limited cluster aggregation) kinetics; liquid-gas separation is prevented by the strength of the bonds, which do not allow restructuration. Good qualitative and quantitative agreement is found between the dynamic simulations and the kinetic model in all the three regions.     

Simplified gauge-cell method and its application to the study of capillary phase transition of propane in carbon nanotubes

A modification of the gauge-cell method for Monte Carlo studies of phase equilibrium in nano-confined systems is presented and employed for studying the capillary phase transition of propane in single-walled carbon nanotubes as a function of tube diameter. It is shown that if an analytical equation of state for the vapor phase is known, the acceptance rule for the trial move of particle exchange can be modified to reference the bulk system through its chemical potential. Under these conditions, the simulation procedure is simplified and acquires many characteristics of a Monte Carlo simulation conducted in the grand canonical ensemble. It is also shown that the critical temperature can be estimated by interpolation of sub- and supercritical values of the slope of the inverse isotherm at the inflection point. To enhance the sampling of propane molecules in the hollow space of the nanotubes the configurational-bias scheme is employed. The simulation results show that the confinement of propane increases its critical density, reduces the critical temperature and narrows the binodal curve with decreasing tube diameter until the system approaches one-dimensional behavior.     

Vapor-liquid interfacial properties of rigid-linear Lennard-Jones chains

We have obtained the interfacial properties of short rigid-linear chains formed from tangentially bonded Lennard-Jones monomeric units from direct simulation of the vapour-liquid interface. The full long-range tails of the potential are accounted for by means of an improved version of the inhomogeneous long-range corrections of Janec?ek [J. Phys. Chem. B 110, 6264-6269 (2006)] proposed recently by MacDowell and Blas [J. Chem. Phys. 131, 074705 (2009)] valid for spherical as well as for rigid and flexible molecular systems. Three different model systems comprising of 3, 4, and 5 monomers per molecule are considered. The simulations are performed in the canonical ensemble, and the vapor-liquid interfacial tension is evaluated using the test-area method. In addition to the surface tension, we also obtain density profiles, coexistence densities, critical temperature and density, and interfacial thickness as functions of temperature, paying particular attention to the effect of the chain length and rigidity on these properties. According to our results, the main effect of increasing the chain length (at fixed temperature) is to sharpen the vapor-liquid interface and to increase the width of the biphasic coexistence region. As a result, the interfacial thickness decreases and the surface tension increases as the molecular chains get longer. The surface tension has been scaled by critical properties and represented as a function of the difference between coexistence densities relative to the critical density.     

Linear aggregation revisited: Rods,rings and worms

The problem of ring formation in solutions of cylindrical micelles is reinvestigated theoretically, taking into account a finite bending rigidity of the self‐assembled linear objects. Transitions between three regimes are found when the scission energy is sufficiently large. At very low densities only spherical and very short, rod‐like micelles form. Beyond a critical density, mainly rings but also worm‐like chains appear in (virtually) fixed relative amounts. Above a second transition both the length of the linear chains and the relative amount of material taken up by them increase rapidly with increasing concentration. The mass accumulated into long, semi‐flexible worms then overwhelms that in rings. The ring‐dominated regime is very narrow for semi‐flexible chains, confirming that the presence of rings may be difficult to observe in many micellar systems, and indeed disappears completely for sufficiently low scission energy and/or large persistence length.