Dynamics of hyperbranched polymers and dendrimers: theoretical models

Chemical reactions depend in many ways on the dynamics of the underlying reactants, and an important aspect is the distance covered by the reactants before the reaction act occurs. Hence, even diffusion-limited reactions between point particles in confined geometries and in low-dimensional systems display decay forms which are very different from those obtained from simple chemical kinetics. Clearly, even more complex decay forms hold for macromolecules, given their internal degrees of freedom. Here we discuss how the dynamics of macromolecules in solution relates to their topological structure and focus on the motion of macromolecular segments (monomers) under the influence of external fields. After a general survey of the method of generalized Gaussian structures (GGS) we recall the wealth of forms which are observed, depending on the topology and on the microscopic dynamics involved. Paradigmatic are the findings for the class of hyperbranched macromolecules; to these belong the dendrimers. While the dendrimers do not show a typical scaling behavior, as found, say, for linear chains, the situation is different for particular classes of regular hyperbranched polymers which are fractal. We end by discussing their pattern of motion in the GGS-Rouse-Zimm picture.     

Dynamics of phase separation and critical phenomena in polymer mixtures

The phenomenological mean-field theory for statics and dynamics of polymer mixtures is described, generalizing the approaches of Flory-Huggins, Cahn-Hilliard and de Gennes. Predictions are made for critical behavior, spinodal decomposition and homogeneous nucleation. The validity of the mean-field approximations is discussed with Ginzburg criteria. The results of the theory are compared to computer simulations and recent experiments.Invited talk delivered at the Tagung der Deutschen Physikalischen Gesellschaft, Fachausschu\ Polymerphysik, Kaiserslautern, 12–14 March 1986, and at the Gordon Research Conference on Polymer Physics, Andover, New Hampshire, 14–18 July 1986.The author is grateful to Dieter W. Heermann, Alla Sariban, Harry L. Frisch, Josef JÄckie and Thomas Schichtel for their fruitful collaboration on this research described in this review. He thanks them and Arthur BaumgÄrtner for allowing to present partially unpublished material, and for stimulating discussions. Furthermore the author has benefitted from discussions with P. G. de Gennes, P. Pincus, H. Sillescu and G. R. Strobl. This research is supported in part by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 41.     

Interfaces between coexisting phases of polymer mixtures: Comparison between Monte Carlo simulations and theoretical predictions

Large scale Monte Carlo investigations of the interface between A-rich and B-rich phases of symmetric binary (AB) polymer mixtures are presented, using the bond fluctuation model of flexible chains with N_A=N_B=N=32 effective monomers. The temperature range studied, 0.144<T/T_c0.759, includes both the strong and the weak segregation limit. Interfacial free energy and interfacial structure are studied, and compared to predictions based on the selfconsistent field theory. Also the broadening of the interfacial width due to capillary waves is considered, and finite size effects due to the confinement of interfaces in thin films of polymer blends are discussed.     

Evolving projection analysis of multicomponent mixtures

The application of evolving projection analysis (EPA) to second-order bilinear data sets consisting of more than two components is described. EPA is a method for rank analysis of ordered data matrices where the components appear sequentially as a function of time or some other ordinal variable. It was found that extension of the method to mixtures of more than two components was best accomplished using principal components analysis to preprocess the data. The algorithm is demonstrated using simulated four-component chromatographic data, and experimental data from liquid chromatography (three- and four-component mixtures) and a spectrophotometric titration (four components), both employing UV-visible diode array detection.     

Optical rheometry of multicomponent polymer liquids

Recent advances in optical rheometric techniques are described, and applications of the methods are presented that motivate their use in elucidating the structure and dynamics of complex, polymeric liquids. Spectroscopic methods, such as infrared dichroism and 2-dimensional Raman scattering, which provide information concerning bond-level orientation dynamics are presented. In addition, the combination of birefringence and stress measurements is demonstrated to be capable of extracting the component dynamics in some multicomponent systems. Examples of the use of these methods include orientation in block copolymers and compatible blends.     

Dynamics of water in partially crystallized polymer/water mixtures studied by dielectric spectroscopy

The dielectric relaxation process of water was investigated for polymer/water mixtures containing poly(vinyl methyl ether), poly(ethyleneimine), poly(vinyl alcohol), and poly(vinylpyrrolidone) with a polymer concentration of up to 40 wt % at frequencies between 10 MHz and 10 GHz in subzero temperatures down to -55 degrees C. These polymer/water mixtures have a crystallization temperature TC of water at -10 to -2 degrees C. Below TC, part of the water crystallized and another part of the water, uncrystallized water (UCW), remained in a liquid state with the polymer in an uncrystallized phase. The dielectric relaxation process of UCW was observed, and reliable dielectric relaxation parameters of UCW were obtained at temperatures of -26 to -2 degrees C. At TC, the relaxation strength, relaxation time, and relaxation time distribution change abruptly, and their subsequent changes with decreasing temperature are larger than those above TC. The relaxation strength of UCW decreases, and the relaxation time and dynamic heterogeneity (distribution of relaxation time) increase with decreasing temperature. These large temperature dependences below TC can be explained by the increase in polymer concentration in the uncrystallized phase C(p,UCP) with decreasing temperature. C(p,UCP) is independent of the initial polymer concentration. In contrast to the relaxation times above TC, which vary with the chemical structure of the polymer and its concentration, the relaxation times of UCW are independent of both of them. This indicates that the factor determining whether the water forms ice crystals or stays as UCW is the mobility of the water molecules.     

Partial miscibility in multicomponent polymer system

It has been shown, using the significant structure theory of liquids, that a lower critical solution temperature behavior as well as a upper critical solution temperature behavior can be expected for polymer–polymer systems and that a phase diagram of closed-loop-type in a polymer–polymer–solvent system can be possible. In this article the sublimation energy of a mixture was expressed as a quadratic form of segment surface fractions on pure components rather than that of mole fractions, and the effect of the segment surface fractions on critical compositions was explained. The calculated partial miscibilities were in good agreement with the experiment.     

Analyzing the composition-property diagrams of multicomponent liquid mixtures

We consider the local regularities of the composition-intensive/specific property diagrams of multicomponent mixtures in which composition is a scalar function of a vector argument. A topographical method used in differential geometry is used to produce one-dimensional, two-dimensional, and three-dimensional scalar fields. The scalar field gradient is used to study the composition-property diagrams of five-component mixtures (producing four-dimensional scalar fields).     

Selective condensation during relaxation of multicomponent gas-plasma mixtures

The physical and chemical processes taking place in multicomponent gas-plasma mixtures are considered. A theoretical model of selective condensation during relaxation of the mixture is developed. It is shown that the condensation process depends strongly on the concentration of charged particles, which appear to be condensation nuclei. The concentration of the charged nuclei isfound to be influenced by the initial conditions of gas-plasma mixture relaxation. Illustrative calculations for titanium ultrafine powder deposition during plasma-chemical processing of titanium tetrachloride are carried out.     

Verification of a theoretical model for multicomponent condensation

In 1974, Taitel and Tamir presented a fully predictive model for the filmwise condensation of multicomponent mixtures. In the present work the predictions of the theoretical model were successfully verified. This was achieved by condensing several binary mixtures (acetone-methanol, acetone-ethanol, methanol-ethanol) and a ternary mixture (acetone-methanol-ethanol) onto a cooled vertical wall.     

Prediction of latent heat of vaporization of multicomponent mixtures

A very simple method has been developed for predicting the differential latent heat of vaporization involved in vapor-liquid equilibria of multicomponent mixtures. The technique uses the UNIFAC method for predicting activity coefficients.In order to establish this method, 88 binary systems and 13 ternary systems, both nonazeotropic and azeotropic, were tested successfully. On the basis of the 101 systems investigated, the mean overall deviation between the observed and predicted values was found to be 4.6%. Despite the fact that the majority of the systems tested were azeotropic, a consequence of the small amount of data available for the differential heats of nonazeotropic systems, the method proposed is equally applicable to nonazeotropic systems.For azeotropic mixtures, it is possible to predict the latent heat (integral or differential) by means of an analytical equation which involves only the parameters of Antoine's equation, with a mean overall deviation of 6.1%.From the differential heats obtained by the proposed method, it is possible to calculate integral heats by applying equations derived here which relate these quantities.     

Liquid-liquid phase separation in multicomponent polymer solutions

Summary In conformity with a prediction byTompa, polymer solutions were found which, in a limited range of concentrations and temperatures, separate into three liquid phases. This was demonstrated with solutions in diphenylether of two polyethylene fractions with narrow molecular weight distributions, which represented the closest possible approximations of ternary systems. The phenomenon takes place exactly under the conditions indicated byTompa on the basis of theFlory-Huggins free enthalpy of mixing function.

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Zusammenfassung In Übereinstimmung mit einer Voraussage vonTompa ließen sich polymere Lösungen finden, die sich in einen begrenzten Bereich von Konzentration und Temperature in drei flüssige Phasen trennen. Dies wurde an Lösungen von zwei Polyäthylenfraktionen mit schmaler M-G-Verteilung in Diphenyläther gezeigt, die die engste mögliche Annäherung an ternäre Systeme repräsentieren. Das Phänomen tritt exakt unter den Bedingungen auf, dieTompa auf Grundlage derFlory-Hugginsschen Freien Mischungsenthalpie angegeben hat.

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With 9 figures in 26 details and 3 tables     

Target factor analysis of infrared spectra of multicomponent mixtures

The target factor analysis (t.f.a.) error criteria discussed earlier are applied to Fourier-transform infrared spectra in order to determine, without any prior knowledge of the experimental error, the number and identities of components in a series of related multicomponent mixtures. The t.f.a. method is also used for quantitative analysis and is compared to regression analysis, which force-fits the data and does not compensate for an impurity of small but measurable quantity. In addition, it is shown how an impurity in a single mixture can be detected, identified, and quantified. An actual experimental example is presented, using solutions of o-xylene, m-xylene, p-xylene, and ethylbenzene in cyclohexane, with chloroform as an impurity.     

Transient separation of multicomponent liquid mixtures in thermogravitational columns

Transient separation of the individual components in a multicomponent liquid mixture in a thermogravitational column can be used to determine the thermal and molecular diffusion coefficients. Two models of the transient behavior are developed. First, a classical model where density only depends on temperature. Second, a general model where the compositional effect on density is also taken into account. Diffusion coefficients can be determined by fitting experimental data to either model. The procedure is demonstrated for a ternary liquid mixture. The results reveal that the classical model is very unreliable even though composition variation only contributes to the total buoyancy by one-third. Diffusion coefficients can be obtained reliably from the general model provided the experimental noise does not exceed +/-1% of steady-state separation. This level of accuracy in composition measurements is achievable.     

On measurement of thermal diffusion coefficients in multicomponent mixtures

We investigate the steady-state separation of the individual components of an incompressible multicomponent liquid mixture in a narrow two-dimensional thermogravitational column. Analytic working equations for measuring thermal diffusion coefficients analogous to the existing equations for a binary mixture are derived. Similar to the binary results, we find that when compositional variation has negligible effect on fluid density and vertical diffusive flux can be ignored, molecular diffusion does not affect steady-state separation. However, when compositional effects on density are taken into account, molecular diffusion does affect the bulk convective flow and the steady-state separation of the components. There may be also two distinct trends in the velocity and separation profiles. With one or more negative thermal diffusion coefficients, there may be more than one convection cell resulting in oscillatory behavior of separation. The working equations presented can be used to measure thermal diffusion coefficients of multicomponent mixtures. Such measurements have not yet been reported in the literature.