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.     

Structures and adsorption of binary hard-core Yukawa mixtures in a slitlike pore: grand canonical Monte Carlo simulation and density-functional study

The grand canonical ensemble Monte Carlo simulation and density-functional theory are applied to calculate the structures, local mole fractions, and adsorption isotherms of binary hard-core Yukawa mixtures in a slitlike pore as well as the radial distribution functions of bulk mixtures. The excess Helmholtz energy functional is a combination of the modified fundamental measure theory of Yu and Wu [J. Chem. Phys. 117, 10156 (2002)] for the hard-core contribution and a corrected mean-field theory for the attractive contribution. A comparison of the theoretical results with the results from the Monte Carlo simulations shows that the corrected theory improves the density profiles of binary hard-core Yukawa mixtures in the vicinity of contact over the original mean-field theory. Both the present corrected theory and the simulations suggest that depletion and desorption occur at low temperature, and the local segregation can be observed in most cases. For binary mixtures in the hard slitlike pore, the present corrected theory predicts more accurate surface excesses than the original one does, while in the case of the attractive pore, no improvement is found in the prediction of a surface excess of the smaller molecule.     

A density functional theory for Lennard-Jones fluids in cylindrical pores and its applications to adsorption of nitrogen on MCM-41 materials

A density functional theory (DFT) constructed from the modified fundamental-measure theory and the modified Benedict-Webb-Rubin equation of state is presented. The Helmholtz free energy functional due to attractive interaction is expressed as a functional of attractive weighted-density in which the weight function is a mean-field-like type. An obvious advantage of the present theory is that it reproduces accurate bulk properties such as chemical potential, bulk pressure, vapor-liquid interfacial tension, and so forth when compared with molecular simulations and experiments with the same set of molecular parameters. Capabilities of the present DFT are demonstrated by its applicability to adsorption of argon and nitrogen on, respectively, a model cylindrical pore and mesoporous MCM-41 materials. Comparison of the theoretical results of argon in the model cylindrical pore with those from the newly published molecular simulations indicates that the present DFT predicts accurate average densities in the pore, slightly overestimates the pore pressure, and correctly describes the effect of the fluid-pore wall interaction on average densities and pressures in the pore. Application to adsorption of nitrogen on MCM-41 at 77.4 K shows that the present DFT predicts density profiles and adsorption isotherms in good agreement with those from molecular simulations and experiments. In contrast, the hysteresis loop of adsorption calculated from the mean-field theory shifts toward the low pressure region because a low bulk saturated pressure is produced from the mean-field equation of state. The present DFT offers a good way to describe the adsorption isotherms of porous materials as a function of temperature and pressure.     

基于密度泛函理论研究二元排斥Yukawa流体的表面结构性质

基于自由能密度泛函理论(DFT)考察了二元排斥Yukawa (HCRY)流体在不同外场下的密度分布. 基于微扰理论, 体系的Helmholtz自由能泛函采用硬球排斥部分和长程色散部分贡献之和, 其中Kierlik和Rosinberg的加权密度近似(WDA)被用来计算硬球排斥部分, 而色散部分采用平均场理论(MFT)进行描述. 为了验证DFT计算结果的合理性, 研究中采用巨正则Monte Carlo(GCMC)模拟计算了在不同主体相密度、硬核直径和位能参数比的条件下二元HCRY混合流体的密度分布. 结果表明, 该DFT计算结果与GCMC模拟值吻合良好.     

Molecular simulation of binary mixture adsorption in buckytubes and MCM-41

MCM-41 and buckytubes are novel porous materials with controllable pore sizes and narrow pore size distributions. Buckytubes are carbon tubes with internal diameters in the range 1–5 urn. The structure of each tube is thought to be similar to one or more graphite sheets rolled up in a helical manner. MCM-41 is one member of a new family of highly uniform mesoporous silicate materials produced by Mobil, whose pore size can be accurately controlled in the range 1.5–10 nm. We present grand canonical Monte Carlo (GCMC) simulations of single fluid and binary mixture adsorption in a model buckytube, and nonlocal density functional theory (DFT) calculations of trace pollutant separation in a range of buckytubes and MCM-41 pores. Three adsorbed fluids are considered; methane, nitrogen and propane. The GCMC studies show that the more strongly adsorbed pure fluid is adsorbed preferentially from an equimolar binary mixture. Ideal adsorbed solution theory (IAST) is shown to give good qualitative agreement with GCMC when predicting binary mixture separations. The DFT results demonstrate the very large increases in trace pollutant separation that can be achieved by tuning the pore size, structure, temperature and pressure of the MCM-41 and buckytube adsorbent systems to their optimal values.     

How critical fluctuations influence adsorption properties of a van der Waals fluid onto a spherical colloidal particle

A recently proposed 3rd-order thermodynamic perturbation theory (TPT) is extended to its 5th-order version and non-uniform counterpart by supplementing with density functional theory (DFT) and a number of ansatzs for a bulk 2nd-order direct correlation function (DCF). Employment of the ansatzs DCF enables the resultant non-uniform formalism devoid of any adjustable parameter and free from numerically solving an Ornstein–Zernike integral equation theory. Density profiles calculated by the present non-uniform formalism for a hard core attractive Yukawa (HCAY) fluid near a spherical geometry are favorably compared with corresponding simulation data available in literature, and are more accurate than those based on a previous 3rd + 2nd-order perturbation DFT. The non-uniform 5th-order TPT is employed to investigate adsorption of the HCAY fluid onto a colloidal particle; it is disclosed that a depletion adsorption can be induced when the coexistence bulk fluid is situated in neighborhood of a critical point or near a bulk vapor–liquid coexistence gaseous phase or liquid phase density. A physical interpretation is given for such depletion adsorption and for its connection with parameters of the potential under consideration, which is ascribed to critical density fluctuations existing within a wide region of the bulk diagram. For a large spherical external potential inducing wetting transition, it is found that only round wetting transition is found instead of 1st-order pre-wetting transition in the case of a planar wall external potential, and the wetting transition temperature increases relative to that for the planar wall external potential. The present theoretical results for wetting transitions are supported by previous investigation based on thermodynamic considerations and a phenomenological Landau mean field theory, and are also in conformity with the present qualitative physical interpretation.     

密度泛函与分子模拟计算介孔孔径分布比较

用巨正则系综Monte Carlo模拟(GCMC)方法和密度泛函理论（ DFT）结合统计积分方程（SIE）计算了介孔材料的孔径分布.为比较这两种方法,以77 K氮气在介孔活性碳微球中的吸附数据为依据,求出其孔径分布.在GCMC模拟和DFT计算中,流体分子模型化为单点的Lerrnard-Jones球;流体分子与吸附剂材料之间的作用采用平均场理论中的10-4-3模型.在DFT方法中,自由能采用Tarazona 提出的加权近似密度泛函方法(weighted density approximation,WDA)求解.结果表明,对于孔径大于1.125 nm的介孔材料,GCMC和DFT两种方法都可以用来研究介孔材料的孔径分布;对于小于1.125 nm的介孔材料,不能用DFT方法计算孔径分布（DFT方法本身的近似产生了误差）,只能用分子模拟方法.     

Density functional theory of inhomogeneous liquids. II. A fundamental measure approach

Previously, it has been shown that the direct correlation function for a Lennard-Jones fluid could be modeled by a sum of that for hard-spheres, a mean-field tail, and a simple linear correction in the core region constructed so as to reproduce the (known) bulk equation of state of the fluid [Lutsko, J. Chem. Phys. 127, 054701 (2007)]. Here, this model is combined with ideas from the fundamental measure theory to construct a density functional theory for the free energy. The theory is shown to accurately describe a range of inhomogeneous conditions including the liquid vapor interface, the fluid in contact with a hard wall, and a fluid confined in a slit pore. The theory gives quantitatively accurate predictions for the surface tension, including its dependence on the potential cutoff. It also obeys two important exact conditions: That relating the direct correlation function to the functional derivative of the free energy with respect to density and the wall theorem.     

Storage of hydrogen at 303 K in graphite slitlike pores from grand canonical Monte Carlo simulation

Grand canonical Monte Carlo (GCMC) simulations were used for the modeling of the hydrogen adsorption in idealized graphite slitlike pores. In all simulations, quantum effects were included through the Feynman and Hibbs second-order effective potential. The simulated surface excess isotherms of hydrogen were used for the determination of the total hydrogen storage, density of hydrogen in graphite slitlike pores, distribution of pore sizes and volumes, enthalpy of adsorption per mole, total surface area, total pore volume, and average pore size of pitch-based activated carbon fibers. Combining experimental results with simulations reveals that the density of hydrogen in graphite slitlike pores at 303 K does not exceed 0.014 g/cm(3), that is, 21% of the liquid-hydrogen density at the triple point. The optimal pore size for the storage of hydrogen at 303 K in the considered pore geometry depends on the pressure of storage. For lower storage pressures, p < 30MPa, the optimal pore width is equal to a 2.2 collision diameter of hydrogen (i.e., 0.65 nm), whereas, for p congruent with 50MPa, the pore width is equal to an approximately 7.2 collision diameter of hydrogen (i.e., 2.13 nm). For the wider pores, that is, the pore width exceeds a 7.2 collision diameter of hydrogen, the surface excess of hydrogen adsorption is constant. The importance of quantum effects is recognized in narrow graphite slitlike pores in the whole range of the hydrogen pressure as well as in wider ones at high pressures of bulk hydrogen. The enthalpies of adsorption per mole for the considered carbonaceous materials are practically constant with hydrogen loading and vary within the narrow range q(st) congruent with 7.28-7.85 kJ/mol. Our systematic study of hydrogen adsorption at 303 K in graphite slitlike pores gives deep insight into the timely problem of hydrogen storage as the most promising source of clean energy. The calculated maximum storage of hydrogen is equal to approximately 1.4 wt %, which is far from the United States Department of Energy (DOE) target (i.e., 6.5 wt %), thus concluding that the total storage amount of hydrogen obtained at 303 K in graphite slitlike pores of carbon fibers is not sufficient yet.     

Structure of inhomogeneous attractive and repulsive hard-core yukawa fluid: grand canonical Monte Carlo simulation and density functional theory study

A density functional theory is proposed for an inhomogeneous hard-core Yukawa (HCY) fluid based on Rosenfeld's perturbative method. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-core repulsion and a quadratic functional Taylor expansion for the long-ranged attractive or repulsive interactions. To test the established theory, grand canonical ensemble Monte Carlo simulations are carried out to simulate the density profiles of attractive and repulsive HCY fluid near a wall. Comparison with the results from the Monte Carlo simulations shows that the present density functional theory gives accurate density profiles for both attractive and repulsive HCY fluid near a wall. Both the present theory and simulations suggest that there is depletion for attractive HCY fluid at low temperature, but no depletion is found for repulsive HCY fluid. The calculated results indicate that the present density functional theory is better than those of the modified version of the Lovett-Mou-Buff-Wertheim and other density functional theories. The present theory is simple in form and computationally efficient. It predicts accurate radial distribution functions of both attractive and repulsive HCY fluid except for the repulsive case at high density, where the theory overestimates the radial distribution function in the vicinity of contact.     

Further test of third order + second-order perturbation DFT approach: hard core repulsive Yukawa fluid subjected to diverse external fields

Grand canonical Monte Carlo simulation is used to investigate density profiles of hard-core repulsive Yukawa (HCRY) model fluid under the influence of various external fields and radial distribution function (RDF) of the bulk HCRY system. The aim of these extensive simulations is to provide exact data for purely repulsive interaction potential against which the validity of a third order + second-order perturbation DFT approach can be tested. It is found that a semiempirical parametrized bridge function due to Malijevsky and Labik performs very well for the RDF of the bulk HCRY fluid. Incorporation of a bulk second-order direct correlation function (DCF) of the HCRY fluid based on the Malijevsky-Labik bridge function into the third order + second-order perturbation DFT approach yields the resulting theoretical predictions for the density profiles of inhomogeneous HCRY fluid that are in a very good agreement with the simulation data, an exception being somewhat larger deviations appearing for the structure of the fluid around the center of a hard spherical cavity. Both theory and simulation predict layering transition and gas-liquid coexistence phenomena occurring with the HCRY model fluid under confined conditions. For the case of an inverse sixth-power repulsive potential under the influence of a flat stationary wall defined by an inverse twelfth-power repulsive potential, the present third order + second-order perturbation DFT approach is found to be superior to several existing weighted density approximations (WDA) and partitioned WDA.     

Structure of inhomogeneous polymer solutions: a density functional approach

The structure of polymer solutions confined between surfaces is studied using a density functional theory where the polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres. The present theory uses the concept of universality of the free energy density functional to obtain the first-order direct correlation function of the nonuniform system from that of the corresponding uniform system, calculated through the Verlet-modified type bridge function. The uniform bulk fluid direct correlation function required as input has been calculated from the reference interaction site model integral equation theory using the Percus-Yevick closure relation. The calculated results on the density profiles of the polymer as well as the solvent are shown to compare well with computer simulation results.     

高阶FMSA展开研究缔合流体界面张力

结合一阶平均球近似(First-order mean-spherical approximation, FMSA)与重整化群(Renormalization group, RG)变换计算了流体全局性相行为. 应用FMSA理论解析得到的径向分布函数(Radial distribution function, RDF)和直接相关函数(Direct correction function, DCF)建立密度泛函方法, 并在其展开项中考虑了高阶微扰项作用, 即考虑了主体流体密度不一致性, 避免原有方法在计算密度分布时存在难以收敛、误差大等问题. 将高阶展开扩展应用到缔合流体, 计算表明, 和分子模拟数据相比, 界面密度分布和界面张力较之原有的密度泛函方法均有了明显改善.     

Sedimentation equilibrium of colloidal suspensions in a planar pore based on density functional theory and the hard-core attractive Yukawa model

The sedimentation equilibrium of colloidal suspensions modeled by hard-core attractive Yukawa (HCAY) fluids in a planar pore is studied. The density profile of the HCAY fluid in a gravitational field and its distribution between the pore and uniform phases are investigated by a density functional theory (DFT) approach, which results from employing a recently proposed parameter-free version of the Lagrangian theorem-based density functional approximation (Zhou, S. Phys. Lett. A 2003, 319, 279) for hard-sphere fluids to the hard-core part of the HCAY fluid, and the second-order functional perturbation expansion approximation to the tail part as was done in a recent partitioned density functional approximation (Zhou, S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 061201). The resultant DFT approach is, thus, the first adjustable parameter-free DFT for HCAY fluids. The validity of the present DFT for HCAY fluids of reduced range parameter z(red) = 1.8 under various external potentials is established in the first of the papers cited previously. The present DFT for HCAY fluids can predict the radial distribution function for the bulk HCAY fluid accurately in the colloidal limit (large value of z(red)), and in the hard-sphere limit, its prediction for the density profile of the hard-sphere fluid in a gravitational field is in very good agreement with the existing simulation data. The dependence of the density profile and distribution coefficient on the magnitude of the interparticle attraction, gravitational field, and degree of confinement is investigated in detail by the present DFT approach. Intuitive and qualitative analyses are also compared with the quantitative DFT calculational results.     

Free energy density functional for adsorption of fluids in nanopores

A classical free energy density functional, which is isomorphic to a usual effective hard sphere model + mean field approximation for tail contribution, is proposed for treatment of real fluids in inhomogeneous states. In the framework of the classical density functional theory (DFT), the present functional is applied to two representative model fluids, namely, a Lennard-Jones fluid and a hard core attractive Yukawa fluid, subject to influence of various external fields. A comprehensive comparison with simulation results and a detailed analysis show that the present functional holds simultaneously all of the desirable properties inherent in an excellent functional, such as high accuracy, computational simplicity, consistency with a hard wall sum rule, nonrecourse to use of adjustable parameter(s) and weighted densities, reproduction of bulk second-order direct correlation function (DCF) in bulk limit, and applicability to subcritical fluid phenomena.     

Bulk and interfacial properties of binary polymer mixtures

A microscopic density functional theory is used to investigate a binary mixture of polymers, built of freely jointed tangent hard spheres. The difference in the chain length and in the segment diameter of polymers gives rise to a demixing transition. We evaluate the bulk fluid phase equilibria (binodal) and the limit of stability of a mixed state (spinodal) for selected systems, and analyze the decay of the critical packing fraction, critical mole fraction, and critical pressure with an increase of the chain length. The bulk results are subsequently used in the calculations of the density profiles across the fluid-fluid interface. The obtained profiles are smooth and do not exhibit any oscillations on the length scale of the segment diameter. Upon approaching the critical point the interfacial tension vanishes as (Deltarho)3, where Deltarho is the difference between bulk densities of one component in bulk phases rich and poor in that species. This indicates that the microscopic density functional theory applied here is of a mean-field type.     

Microscopic structure and properties of an interface between coexisting phases of an associating fluid adsorbed in slitlike pores

Using density functional theory, we calculate density profiles of an associating fluid in slitlike pores. These profiles characterize an interface between two coexisting, adsorbed phases, e.g., between gaseous and liquid phases formed during capillary condensation. Our study has been carried out for weakly, as well as for strongly, associated fluids confined in pores of different widths. We also investigate the role of the fluid-wall interaction.     

Analysis of the validity of perturbation density functional theory: based on extensive simulation for simple fluid at supercritical and subcritical temperature under various external potentials

Because of the scarcity of available simulation data for confined hard-core attractive Yukawa model fluid, extensive Monte Carlo (MC) simulation research for this fluid under the influence of various external potentials were carried out. The present MC simulation results were employed to test a performance of the third-order perturbation density functional theory (DFT) based on a high order direct correlation function (DCF) [S. Zhou and E. Ruckenstein, Phys. Rev. E. 61, 2704 (2000)]. It was found that the present perturbation DFT formalism is soundly structured only if the imported second-order DCF is reliable. In this case, the accuracy of the results can be satisfactory or even very high for various types of external potentials. Further, the associated adjustable parameter can be universal, i.e., independent of the particular external field responsible for the generation of a nonuniform density profile. Dependence of both the maintenance of the reliability of the formalism and holding of the universality of the adjustable parameter on the accuracy of the imported bulk second-order DCF can be strengthened by the large difference between the external field investigated and that caused by a single hard wall used for specification of the adjustable parameter. In case the gaseous density in the subcritical region is below the coexistence density, an excellent performance of the present formalism is observed even for the mean spherical approximation's second-order DCF as an input. This advantageous property, combined with the fact that the present formalism needs only the second-order DCF of fluid at the coexistence state as an input, enables the present formalism to be a very good theoretical tool for the investigations of wetting and prewetting transitions.