Edmund A. DiMarzio

Adsorption of a Chain Polymer between Two Plates

A lattice model of adsorption of an isolated chain polymer between two plates is investigated using a matrix formalism and a grand canonical ensemble (GCE) formalism. The matrix formalism is particularly convenient for calculating the polymer segment density as a function of the distance from one of the plates for different fixed plate separations. The GCE formalism can be used to calculate the fraction of loops (sequences of polymer segments whose ends are in contact with one plate and whose intermediate segments lie between the two plates), bridges (sequences of polymer segments whose ends are in contact with different plates and whose intermediate segments lie between the two plates), and trains (sequences of polymer segments which are wholly in contact with one plate or the other). All of the foregoing quantities have been calculated in the limit of infinite molecular weight as a function of the distance of separation between the plates and the energy of adsorption of a polymer segment on a plate. The...

Chain Stiffness and the Lattice Theory of Polymer Phases

General formulas for the thermodynamic properties of amorphous polymer phases are obtained from statistical mechanics, with the aid of the lattice model, in a manner which avoids the use of restrictive assumptions concerning the nature of the individual polymer chains.Certain results, such as prediction of a second‐order transition for systems of semiflexible chains and the Flory‐Huggins formula for the entropy of mixing with monomeric solvent, are thus shown to be independent of the precise nature of the model assumed for the polymer chains.More complete information may be obtained by application of the general formulas to models descriptive of the molecular chains in question. As an example, the results of Flory for semiflexible linear chains whose stiffness arises exclusively from intramolecular nearest neighbors are obtained as a special case. (The conventional thermodynamic properties of polydisperse systems of chains of this type are shown to depend on the number average molecular weight.)

Statistics of Orientation Effects in Linear Polymer Molecules

This paper is concerned with the effects of orientation on the combinatorial term g for the number of ways to pack together Nx linear polymers (x mers). Accordingly g is evaluated as a function of the number of molecules in each permitted direction for the case of straight rigid rods. The permitted directions can be continuous so that g is derived as a function of the continuous function f(r) which gives the density of rods lying in the solid angle Δr, or the permitted directions can be discrete so that g is the number of ways to pack molecules onto a lattice. To illustrate the usefulness of the orientation dependent combinatorial terms, liquid crystals are discussed. Another phase is found to exist in addition to the previously predicted nematic phase. This phase is tentatively identified with the cholesteric phase.A procedure is developed for the calculation of the orientation dependent combinatorial term associated with the packing together of molecules of arbitrary shape. A very approximate applicatio...

Proper Accounting of Conformations of a Polymer Near a Surface

It is well known that in free space the conformations of a freely jointed chain (n‐mer) can be generated with proper equal a priori probabilities by means of a particle performing an n‐step random walk. However, near a boundary the method of a random walk with reflecting barriers weights too heavily those paths that touch the boundary r times by a factor (w)r where w is greater than 1. The essence of a proper accounting is to place a completely absorbing site just beyond the boundary and to count only those chains that do not terminate on the absorbing site. For example, in the limit of large n where the diffusion equation becomes valid, the proper boundary condition is that of complete absorption at the boundary (concentration equals zero) rather than complete reflection (gradient of concentration equals zero) as has been assumed previously. In considering the problem of a polymer confined to a finite length strip of a one‐dimensional lattice one ends up considering nonstochastic submatrices of the matri...

Adsorption of Polymer Molecules at Low Surface Coverage

A statistical‐mechanical treatment of a polymer molecule adsorbed on a solid surface is given. The surface coverage by adsorbed molecules is assumed to be sufficiently low that the interactions of the adsorbed polymer molecules with each other may be neglected. The partition function is derived for a polymer molecule with sequences of repeating units adsorbed at an interface and with other sequences (loops) held at the surface only at their ends. The assumption of Gaussian statistics for the loops leads to a formulation equivalent to that used for the helix‐coil region in DNA molecules. A broad distribution of loop sizes is found, in contrast to Silberbergs theory in which a sharply peaked distribution is assumed. The latter theory predicts also small loops for all values of the adsorption free energy. In contrast, our theory predicts large loops and few units adsorbed for small adsorption free energies and small loops and more units adsorbed for larger adsorption free energies when the chains are suffic...

One‐Dimensional Model of Polymer Adsorption

A detailed treatment of the conformations of a one‐dimensional polymer molecule adsorbed to a surface is given. The average number of contacts of the chain with the surface, the end‐to‐end length, and the distribution of segments ρ(z) with respect to distance z from the surface are computed as functions of the chain length (N) of the polymer and the attractive energy of the surface. Both theoretical and Monte Carlo calculations are used. A transition is found at an attractive energy of kT ln 2. For attractive energies less than this value, the average number of contacts of the chain with the surface approaches a finite value as N approaches infinity, while the end‐to‐end length vaires as N½. However, above the transition the number of contacts is proportional to N and the end‐to‐end length is independent of N. The distribution of segments ρ(z) also shows a marked change as we go through the transition. The one‐dimensional model is shown to correspond to the projection of a three‐dimensional model on the d...

Dilute Solution Theory of Polymer Crystal Growth: A Kinetic Theory of Chain Folding

A kinetic theory of polymer crystallization from dilute solution is formulated for linear chain molecules of finite molecular weight (monodisperse). Two models of crystal growth are considered; both are essentially “regular” chain folding type models. Formulas for the crystal growth rates are derived as a function of the fundamental rate constants associated with the various states of molecular crystallization. These rate constants are evaluated as a function of polymer concentration, molecular weight, crystallization temperature, and crystal thickness. Consideration of finite molecular weight molecules requires an understanding of how these molecules are incorporated into the crystal and what happens to chain ends. Attention is focused on these problems and a description of how “cilia” are formed in polymer crystals is given. A remarkable aspect of cilia formation is that the uncrystallized portion of a chain molecule which dangles in the solution can participate in nucleating a new growth strip (fold pl...

Modelling the amorphous phase and the fold surface of a semicrystalline polymer—the Gambler's Ruin method

Abstract A semicrystalline polymer with lamellar morphology consists of alternating amorphous and crystalline regions. If sufficiently long, each molecule in this system traverses both the crystalline and amorphous zones. The amorphous portion is comprised of portions of a molecule that form loops that re-enter the same lamella at some distance from the point of emergence, and bridges that form connections between two different crystal lamellae. (A tight fold is not considered to be a loop). The statistics of loops and bridges are shown to be identical to the classical Gamblers Ruin problem in mathematical statistics. This is a useful observation because the extensive existing literature on the Gamblers Ruin problem allows us immediately to transcribe results to the polymer system. Using this approach, the ratio of the number of loops to the number of bridges is determined to be M , the thickness of the amorphous zone in unit statistical steps. Also, the average number of steps comprising the amorphous run is determined to be 3 M +1 for a simple cubic lattice in three dimensions. This modelling leads to a calculation of the minimal fraction of crystal stems involved in tight folding in a semicrystalline polymer. For a simple cubic lattice this is found to be 2 3 . The effects of crystal structure and stiffness of the chain in the melt on this bound are discussed.

Contribution to a Liquid‐Like Theory of Rubber Elasticity

The contribution to the configurational entropy of a rubber arising from the competition of chain segments for space is evaluated and the associated entropy force is derived. This is done by expressing the orientation‐dependent packing entropy as a function of the number of bonds (vector lengths between segments) in each of the permitted directions and then expressing the number of bonds in each direction as a function of the stretch ratios, λi. To make the calculations tractable, the three‐chain model of a rubber is used and in addition only three mutually perpendicular orientations of the bonds are allowed. The predicted stress‐strain curves are compared with the experimental situation. It is found that, for a dry rubber, the correction term has the same behavior at small and moderate elongations as the difference between the experimentally observed curve and the curve predicted by the uncorrected statistical theory. However, the magnitude of the correction term is from 1/10 to ½ the magnitude of the ob...

Theoretical prediction of the specific heat of polymer glasses

Lattice vibrations are incorporated into the Gibbs‐DiMarzio configurational entropy theory of glasses. A comparison is made with data compiled by O’Reilly. The resulting theory predicts the specific‐heat discontinuity at the glass transition to within 20%. No adjustable parameters are involved; only the chemical structure need be known. The specific‐heat discontinuity has three parts. The formula in customary notation is Δcp=R (Δe/kTg)2 f (1−f) +RTgΔα (4−TgΔα/0.06) +0.5TgΔαcp(Tg−). The first part is a configurational term arising from shape changes of the molecules (?50% of total). The second part is a configurational term arising from volume expansion (?30% of total). Finally there is a vibrational contribution arising from the change of the force constants (or characteristics frequencies) with temperature (?20% of total).