Marios K. Kosmas

On scaling theories of polymer solutions

We present a rigorous derivation of a scaling theory of the thermodynamics of polymer solutions at finite concentrations. The derivation proceeds directly in 3(2, 4, etc.) dimensions from the expression for the partition function for a solution of monodisperse continuous Gaussian chains with excluded volume. There is no need for the use of renormalization group methods or for the extrapolation of results from calculations near four‐dimensions. Nethertheless, the importance of four dimensions emerges directly from the scaling theory as does a proof that only the binary excluded volume parameter appears in the equation of state, average chain dimensions, etc., in the limit of long enough chains. The irrelevance of the range of the binary segment interaction and the sufficiency of the two parameter theory is thus established by the scaling theory in a straightforward and elementary fashion. We demonstrate that the power law dependence, ν in 〈R2〉∝L2ν, of the mean square end‐to‐end distance 〈R2〉 on the chain l...

Effects of the excluded volume interactions on the conformational properties of star polymers

Abstract The effects of the excluded volume interactions on the conformational properties of star polymers have been studied. First order calculations at the critical dimensionality d = 4 yield the critical exponents of the average quantities up to first order in e = 4-d. We thus find the partition function, the probability of the end of a branch to reach the central core and the probability of contact of the ends of two branches. The size of the macromolecule, expressed by the mean square radius of gyration 〈s2〉star is studied in the region where the interactions between the polymeric units repel one another and in the region where the units attract one another. The results are compared with the results of previous works and with experiments.

On the transition to the power law dependence of stiff and flexible macromolecules

Abstract By means of a theoretical expression relating the average size of a chain and the characteristics of the interactions between the polymeric units, we explain the two different ways of approach to the power law dependence of flexible and stiff macromolecules.

Configurational properties of dilute polymer solutions, dimensionality and perturbation theory

By using second-order perturbation theory in the small parameter epsilon =4-d to 0, the author determines a specific value of the excluded volume parameter u equivalent to the fixed point value given by renormalisation group theory. For this value of the excluded volume parameter each expansion series in epsilon can be summed to an exponential function. The author thus studies the total number of configurations, C, the number of configurations returning to the origin, U, and the mean square end-to-end distance, (R2), of the polymer coil. An interdimensional relationship developed by Kosmas and Freed (1978) is used to extrapolate the authors results to lower dimensions. Finally, the author compares his results with those of previous theories and lattice enumerations, discussing possible differences between the Gaussian excluded volume model used by him and the self-avoiding walk model, close to dimensionality d=1.

Self‐consistent field theories of the polymer excluded volume problem. IV. The linear polymer

The previous abstract formulation of self‐consistent field (SCF) theories of polymer excluded volume is extended and generalized to provide a SCF theory for the moments of the end‐vector distribution such as 〈R2n〉. The nonlinear integrodifferential equations of the SCF theory are considered for a linear polymer where the SCF’s, V2n, are shown to be spherically symmetric, and the dominant part of V2n is obtained and shown to be self‐consistent. The 〈R2n〉 are obtained and have asymptotic long chain limits conforming to the Flory–Edwards power laws. Contributions from fluctuations about the SCF are considered and are shown to alter the total number of chain configurations but not to change the long chain limits for 〈R2n〉 for n≳0. A discussion is presented of the relationship of the SCF results to those obtained by use of the renormalization group method.

A non-ideal polymer chain interacting with a penetrable surface

The author studies the conformational properties of a polymer chain in the presence of both excluded volume interactions and interactions with a penetrable surface. The coexistence of two different kinds of interaction brings new features to the behaviour of the chain, beyond those coming from the two interactions acting independently. He follows an analysis which can find application to the study of problems with more than one interaction parameter.

Localized and nonlocalized polymer chains interacting with a surface

The conformational properties of a polymer chain interacting with a surface are studied and new features for the cases of a chain free to move in the whole space and for a localized chain fixed with one of its ends at a point are clearly illustrated. The progress achieved is based on exact solutions given in the frame of the continuous model by means of integral equations and Laplace transforms. They permit the study of the behavior of the chain at any distance from the surface as a function of its length, the intensity of the polymer‐surface interactions, and the size L of the available space perpendicular to the surface for the nonlocalized chain or the localization position for the localized chain, respectively. The necessary attractions for the gathering of the nonlocalized chains at the surface are larger for larger freedom of the chains which increases with L . From the dependence on the surface‐polymer interactions, density profiles are determined for both the depletion and adsorption cases. The be...

The effects of substrate interactions in the liquid chromatography of polymers

We present a study of the elution behaviour of polymers through a liquid chromatographic column and its dependence on the interactions with the substrate. An earlier method based on statistical thermodynamics is employed to describe chains in the presence of interacting cubic pores in terms of the probability to find the chains in and far from the pores. The partition coefficient of the chains between the mobile and the stationary phases, as a function of the polymer molecular weight, the average size of the pores and a polymer-substrate interaction parameter, is determined and compared with the results of supportive experiments. Two series of chromatographic experiments with two columns with substrates different in nature but with the same pore size, indisputably shows the importance of the polymer-surface interactions. Due to the change of the nature of the substrate an alternation of the mode of elution of the same poly(methyl methacrylates) in the same solvent is observed.

Conformational properties of regular comb polymers

Regular comb polymers with excluded volume interactions are studied and compared with regular star and linear polymers. First-order calculations in the excluded volume parameter, at the critical dimensionality d=4, yield the characteristic exponents of the macroscopic properties to order epsilon =4-d. The number of total configurations and the number of configurations with the backbone forming a ring are found. The evaluation of the mean end-to-end square distances of the backbone and the branches give an insight to the spatial distribution of the macromolecule.

On the mean radius of gyration of a polymer chain

Calculates the intrachain bead-to-bead square distance of a real polymer chain, close to the critical dimensionality d=4. From this the authors obtain the mean-square radius of gyration (S2) of the polymer as a function of the molecular weight N of the chain and the excluded volume parameter mu . A proportionality relationship between (S2) and the mean square end-to-end distance (R2) of the coil, previously suggested by enumerations of self-avoiding walks on lattice, is shown to be true. An estimate of the universal ratio (S2)/(R2) is given.