Phase coexistence in a polydisperse charged hard-sphere fluid: polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.     

Effects of Polydispersity on the Phase Behavior of Additive Hard Spheres in Solution

The ability to separate enzymes, nucleic acids, cells, and viruses is an important asset in life sciences. This can be realised by using their spontaneous asymmetric partitioning over two macromolecular aqueous phases in equilibrium with one another. Such phases can already form while mixing two different types of macromolecules in water. We investigate the effect of polydispersity of the macromolecules on the two-phase formation. We study theoretically the phase behavior of a model polydisperse system: an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We calculate the theoretical liquid–liquid phase separation boundary (the binodal), the critical point, and the spinodal. We vary the distribution of the polydisperse component in terms of skewness, modality, polydispersity, and number of sub-components. We compare the phase behavior of the polydisperse mixtures with their concomittant monodisperse mixtures. We find that the largest species in the larger (polydisperse) component causes the largest shift in the position of the phase boundary, critical point, and spinodal compared to the binary monodisperse binary mixtures. The polydisperse component also shows fractionation. The smaller species of the polydisperse component favor the phase enriched in the smaller component. This phase also has a higher-volume fraction compared to the monodisperse mixture.     

Numerical study of the phase behavior of rods in solution: Warner–Flory model with polydisperse system of rods

Phase equilibria of polydisperse rods in solution are studied in terms of the Warner–Flory model. Numerical calculations are made with a Gaussian-like asymmetric distribution of rod lengths. The aim of these studies is to investigate (i) the dependence of the phase diagram on the shape of the polydispersity distribution function and (ii) the isotropic–anisotropic phase separation within the coexistence range. It is shown that, as a consequence of polydispersity, even for a narrow distribution, the phase diagram is very simple, with only one type of anisotropic phase present. However, within the biphasic range, the properties of the anisotropic phase vary significantly with temperature. At high temperatures, they are essentially governed by steric effects and at low temperatures by both steric effects and soft anisotropic interactions between the rods.     

Polydispersity effects and the interpretation of polymer adsorption isotherms

In many cases, polymer adsorption is studied by measuring adsorption isotherms. Quite often it is found that the results are at variance with theoretical predictions. However, usually these adsorption isotherms are interpreted in terms of a single polymeric solute. Most polymers used in experimental studies are polydisperse and should be treated as mixtures. It is well established that the larger molecules in such mixtures adsorb preferentially over the smaller ones. In this paper we show that many discrepancies between polymer adsorption theory and experiment (e.g., the rounded shape of isotherms, the dependence of the adsorbance on adsorbent concentration, and the lack of desorption upon dilution) can be attributed to polydispersity. A quantitative analysis enables us to calculate isotherms for a polymer of arbitrary molecular weight distribution, provided the dependency of the plateau adsorbance on molecular weight is known. Experiments supporting the theory are reported. The fact that polymers do not desorb upon dilution with solvent is often regarded as a proof that polymer adsorption is irreversible. We show that, if a polydisperse sample is in equilibrium with an adsorbing surface, no detectable desorption may take place upon dilution. Therefore, the adsorption of polymers might well be reversible, even if desorption experiments would indicate apparent irreversibility.     

Effect of quenched size polydispersity on the fluid-solid transition in charged colloidal suspensions

We study the effect of quenched size polydispersity on the phase behavior of charged colloidal suspensions using free-energy calculations in Monte Carlo simulations. The colloids are assumed to interact with a hard-core repulsive Yukawa (screened-Coulomb) interaction with constant surface potential, so that the particles are polydisperse both in size and charge. In addition, we take the size distribution to be fixed in both the fluid and crystal phase (no size fractionation is allowed). We study the fluid-solid transition for various screening lengths and surface potentials, finding that upon increasing the size polydispersity the freezing transition shifts toward higher packing fractions and the density discontinuity between the two coexisting phases diminishes. Our results provide support for a terminal polydispersity above which the freezing transition disappears.     

Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: high temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere fluid mixtures. It is demonstrated that in the frames of the HTA the model belongs to the class of "truncatable free energy models," i.e., the models with thermodynamical properties (Helmholtz free energy, chemical potential, and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and size distribution functions of the coexisting phases of several different models of polydisperse fluids. In particular, we consider polydisperse one-Yukawa hard-sphere mixture with factorizable Yukawa coefficients and polydisperse Lennard-Jones (LJ) mixture with interaction energy parameter and/or size polydispersity. To validate the accuracy of the HTA we compare theoretical results with previously published results of more advanced mean spherical approximation (MSA) for the one-Yukawa model and with the Monte Carlo (MC) computer simulation results of [Wilding et al. J. Chem. Phys. 121, 6887 (2004); Phys. Rev. Lett. 95, 155701 (2005)] for the LJ model. We find that overall predictions of the HTA are in reasonable agreement with predictions of the MSA and MC, with the accuracy range from semiquantitative (for the phase diagram) to quantitative (for the size distribution functions).     

“Salt in a porous matrix” adsorbents: Design of the phase composition and sorption properties

This review formulates the concept of target-oriented synthesis of two-component “salt in a porous matrix” (SPM) adsorbents designed for processes such as gas dewatering, moisture control, heat conversion in adsorption heat pumps, and equilibrium shifting in catalytic reactions. In terms of this approach, the requirements imposed on an ideal adsorbent, which is optimal for a particular application, are initially formulated; then, a material with nearly optimal properties is synthesized. Methods for the target-oriented synthesis of SPM adsorbents with the required properties are considered. The effects of the nature of the salt and the matrix, the salt content, the pore size of the matrix, and the synthesis conditions on the phase composition and adsorption properties of the SPM adsorbents are studied.     

Phase behavior of weakly polydisperse sticky hard spheres: perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys. 49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.     

Elasticity of two-dimensional crystals of polydisperse hard disks near close packing: surprising behavior of the Poisson's ratio

The equation of state, elastic constants, and Poisson's ratio of a crystalline two-dimensional polydisperse hard disk system were determined in the close packing limit. Monte Carlo simulations in the NpT ensemble with variable shape of the periodic box reveal that the pressure and elastic constants grow with increasing polydispersity. The equation of state and the bulk modulus are well described by the free volume approximation. The latter approximation fails, however, for the shear modulus. The simulations also show that the introduction of any amount of size polydispersity in the hard disk systems causes a discontinuous "jump" of the Poisson's ratio in the close packing limit from the value ν(δ=0) = 0.1308(22), obtained for equidiameter hard disks, to ν(δ>0) ≈ 1, estimated for the polydisperse disks.     

Polydisperse hard spheres at a hard wall

The structural properties of polydisperse hard spheres in the presence of a hard wall are investigated via Monte Carlo simulation and density functional theory (DFT). Attention is focused on the local density distribution rho(sigma,z), measuring the number density of particles of diameter sigma at a distance z from the wall. Estimates of rho(sigma,z) are obtained for bulk volume fractions eta(b)=0.2 and eta(b)=0.4, and for two choices of the bulk parent distribution: a top-hat form, which we study for degrees of polydispersity delta=11.5% and delta=40.4%, and a truncated Schulz form having delta=40.7%. Excellent overall agreement is found between the DFT and simulation results, particularly at eta(b)=0.2. A detailed analysis of rho(sigma,z) confirms the presence of oscillatory size segregation effects, as observed in a previous DFT study [I. Pagonabarraga, M. E. Cates, and G. J. Ackland, Phys. Rev. Lett. 84, 911 (2000)]. For large delta, the character of these oscillation is observed to depend strongly on the shape of the parent distribution. In the vicinity of the wall, attractive sigma-dependent depletion interactions are found to greatly enhance the density of the largest particles. The local degree of polydispersity delta(z) is suppressed in this region, while further from the wall it exhibits oscillations.     

Examining the effect of chain length polydispersity on the phase behavior of polymer solutions with the statistical associating fluid theory (Wertheim TPT1) using discrete and continuous distributions

Polymers are naturally polydisperse. Polydispersity may have a large effect on the phase behavior of polymer solutions, in particular, on the liquid-liquid phase equilibria. In this paper, we determine the cloud and shadow curves bounded by lower critical solution temperatures for a number of polymer+solvent systems where the polymer is polydisperse in terms of molecular weight (chain length). The moment method [P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem. Phys. 116, 265 (2001)] is applied with the SAFT approach to determine cloud and shadow curves with continuous Schulz-Flory distributions. It is seen that chain length polydispersity always enhances the extent of liquid-liquid phase equilibria. The predicted cloud curves obtained for continuous distributions are very similar to those obtained for simple ternary mixtures with the same polydispersity index, while the corresponding shadow curves can be very different depending on the composition of the parent distribution. The ternary phase behavior can be used to provide an understanding of the shape of the cloud and shadow curves. Regions of phase equilibria between three liquid phases are found for ternary systems when the chain length distribution is very asymmetrical; such regions are not observed for Schulz-Flory distributions even in the case of a large degree of polydispersity.     

Phase coexistence in polydisperse charged hard-sphere fluids: mean spherical approximation

Taking advantage of the availability of the analytic solution of the mean spherical approximation for a mixture of charged hard spheres with an arbitrary number of components we show that the polydisperse fluid mixture of charged hard spheres belongs to the class of truncatable free energy models, i.e., to those systems where the thermodynamic properties can be represented by a finite number of (generalized) moments of the distribution function that characterizes the mixture. Thus, the formally infinitely many equations that determine the parameters of the two coexisting phases can be mapped onto a system of coupled nonlinear equations in these moments. We present the formalism and demonstrate the power of this approach for two systems; we calculate the full phase diagram in terms of cloud and shadow curves as well as binodals and discuss the distribution functions of the coexisting daughter phases and their charge distributions.     

Particle Penetration into Polydisperse Polymer Brushes: A Theoretical Analysis

Based on theoretical analysis, the effect of polydispersity on particle penetration into polydisperse polymer brushes is investigated. Three different polydispersities representing sharp, moderate, and extremely wide chain length distributions are chosen, since the corresponding explicit expressions of brush density at these polydispersities are available. To simplify the discussion, this study is restricted to spherical particles of small size which ensure that the particle insertion only causes local conformational perturbations. By analyzing the particle distribution, it is found that polydispersity always facilitates particle penetration. This prediction is confirmed by analyzing the surface fluctuations of the brushes. Interestingly, uniform scaling relations are observed for particles penetrating into monodisperse and moderately polydisperse brushes. The uniformity predicted by monodisperse and moderately polydisperse brushes originates from the same asymptotic behavior of their densities approaching the brush edge. This indicates that polydispersity brings significant influence only at high polydispersities.     

Principles of Considering the Effect of the Limited Volume of a System on Its Thermodynamic State

The features of a system with a finite volume that affect its thermodynamic state are considered in comparison to describing small bodies in macroscopic phases. Equations for unary and pair distribution functions are obtained using difference derivatives of a discrete statistical sum. The structure of the equation for the free energy of a system consisting of an ensemble of volume-limited regions with different sizes and a full set of equations describing a macroscopic polydisperse system are discussed. It is found that the equations can be applied to molecular adsorption on small faces of microcrystals, to bound and isolated pores of a polydisperse material, and to describe the spinodal decomposition of a fluid in brief periods of time and high supersaturations of the bulk phase when each local region functions the same on average. It is shown that as the size of a system diminishes, corrections must be introduced for the finiteness of the system volume and fluctuations of the unary and pair distribution functions.     

Investigation of structure factors in dense colloidal suspensions using frequency domain photon migration: polydisperse systems

Frequency domain photon migration (FDPM) technique was employed to investigate the structure factors of dense, polydisperse colloidal suspensions. The angle-integrated structure factors, [S(q)], extracted from FDPM measurements of scattering properties at volume fractions ranging from 0.05 to 0.4, were compared with the values predicted from the polydisperse hard sphere Percus-Yevick (HSPY) model, as well as decoupling approximation (DA) and local monodisperse approximation (LMA) models that incorporated independently measured particle size information. Results show that the polydisperse HSPY model is the most suitable for accounting for particle interactions which predominantly arise from volume exclusion effects. Furthermore, the influence of size polydispersity upon [S(q)] is most significant at high volume fractions. The static structure factors at small wave vector q, S(0), were also assessed from dual wavelength FDPM measurements by using the small wave number approximation as well as the local monodisperse approximation. The measured S(0) agrees well with the values predicted by the polydisperse HSPY model.     

Size fractionation in a phase-separated colloidal fluid

Phase separation of a polydisperse colloidal dispersion implies size fractionation. An application of this effect is given by size-selective purification procedures associated with the colloidal synthesis of so-called monodisperse nanoparticles. We used electron microscopy to determine detailed particle size distributions of coexisting colloidal fluid phases containing highly polydisperse iron oxide nanoparticles with a log-normal distribution (sigma = 0.54 for the total system). Analysis of N approximately 10000 particles per phase yields the first five statistical moments of the distributions. Within experimental error, the interdependence of the statistical moments is in quantitative agreement with the "universal law of fractionation" proposed by Evans, Fairhurst, and Poon [Phys. Rev. Lett. 1998, 81, 1326], even though the theory was derived in the limit of slight polydispersity.     

Study on the concentration effects in size exclusion chromatography VII. A quantitative verification for the model theory of concentration and molecular mass dependences of hydrodynamic volumes for polydisperse polymers

A model theory of concentration effects for polydisperse polymers was proposed in 1988. It is successful in relating the concentration of the injected solution to the effective hydrodynamic volumes of peak, the retention volumes of peak and the polydispersity index (Dc = (Vhcw)/(Vhcn) of hydro-dynamic volume distribution for polydisperse polymers at a given concentration. The dependence of the concentration of injected polymer solution on the effective hydrodynamic volumes, the retention volumes of peak and the polydispersity index of hydrodynamic volume distribution for narrow disperse and polydisperse polystyrene, poly(dodecyl methacrylate), poly(tridecyl methacrylate) and poly(methyl methacrylate) in tetrahydrofuran solvent were studied. The proposed theory was verified by these experimental data. Results show that the proposed theory can predict the concentration effects in GPC for polydisperse polymers quantitatively and can provide a theoretical foundation for the two methods of calibrating the universal calibration curves with polydisperse polymers and of determining the second virial coefficients (A2) of polymers. It is found that the determined values of A2 for narrow disperse and polydisperse polymers by the proposed method are in agreement with those obtained by the LALLS method, and the two universal calibration curves with narrow disperse and polydisperse polymers are in excellent agreement.     

石墨烯基吸附剂的设计及其对水中抗生素的去除

抗生素的大量使用,所带来的环境污染问题受到广泛关注。吸附法因去除效率高、普遍适用性强,呈现出广阔的应用前景,开发新型吸附剂是高效能吸附处理的关键。近年来石墨烯优良的物理和化学性质以及吸附性能,使其成为重要的抗生素吸附剂。由于石墨烯自身的局限性以及对石墨烯吸附剂处理效能和稳定性的要求,基于石墨烯设计开发了多种石墨烯基吸附材料。而目前基于水体中抗生素的石墨烯基复合材料的设计、合成及其吸附作用机制缺乏相关的系统性综述。本文综述了目前水体中抗生素的危害,针对石墨烯基复合吸附材料中,广泛关注的磁性石墨烯吸附剂、聚合物/石墨烯吸附剂、三维石墨烯凝胶和石墨烯/生物炭吸附剂的设计和制备方法进行了总结和概述,并阐述了石墨烯基吸附材料对水体中抗生素的主要吸附作用机制。最后,本文对石墨烯基吸附材料去除水体中抗生素未来的发展方向进行了展望。     

Adsorbents from Waste materials

The possibility of using pyrolyzed wastes produced in already working incineration plants, as adsorbents for waste water treatment, was studied. Showing very poor adsorption properties, they were improved by steam activation technique used in the conventional activated carbon manufacturing. It is concluded that various organic waste materials can be converted to carbonaceous final products with a character similar to activated carbon. Their adsorption properties and pore size distribution are determined by the structure of the starting material. Although most of these samples have a low specific surface area, their pore volume is not negligible in the meso-and micropore range. Adsorption tests with model waste waters confirmed that adsorption properties are strongly influenced by the character of the suface. The adsorption capacity of these samples can be utilized for the treatment of strongly polluted industrial waste waters. Considering that the raw material ‘needed’ to manufacture these adsorbent is produced permanently and the adsorbents do not have to be regenerated, it might be worthwhile using these kinds of adsorbents in the primary treatment of industrial waste waters.