The thermodynamic properties of amorphous phases of linear molecular chains are obtained from statistical mechanics by means of a form of the quasi-lattice theory which allows for chain stiffness and the variation of volume with temperature. A second-order transition is predicted for these systems. This second-order transition has all the qualitative features of the glass transition observed experimentally. It occurs at a temperature which is an increasing function of both chain stiffness and chain length and a decreasing function of free volume. The molecular ``relaxation times'' are shown to increase rapidly as the second-order transition temperature is approached from above. To permit quantitative application of the theory and determine the relationship between the second-order transition and the glass transition observed in ``slow'' experiments these two transitions are tentatively identified. By this means quantitative predictions are made concerning the variations of (1) glass temperature with molecular weight, (2) volume with temperature, (3) volume with molecular weight, (4) volume at the glass temperature with the glass temperature for various molecular weights of the same polymer, (5) specific heat vs temperature, and (6) glass temperature with mole fraction of low-molecular weight solvent, since extensive experimental results are available for these properties. These and other theoretical predictions are found to be in excellent agreement with the experimental results.

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We treat a diblock copolymer of lamellar morphology where one of the blocks is amorphous and one is crystalline (amphiphilic copolymer). The proposed models allow for the stretching of polymer chains, the change in packing entropy arising from changes in orientation of bonds, and the space-filling properties of the chains. Formulas are given for the thickness of the amorphous and crystalline lamellae, Za and Zc, as functions of the lengths of the blocks, ra and rc, the surface and fold free energies, a, and at, the temperature T, the amount of solvent in the amorphous phase v0 = 1vz, and the densities pa and pc (pa = vzp0). We have Za = ra2/3(<78 + vtpjV'HSkTpjW and Zc = r^c, + *,Pc)1/3/Pcra1/3(3m1/3.

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 self-excluded volume of the polymer chain is ignored. In addition the average sizes of loops, bridges, and trains, and the effective force of attraction between the plates is calculated.

Dilute solutions of finite size particles undergoing Brownian motion and flowing through a capillary have average velocities which depend on the particle size. Thus one can obtain a separation of particles of different sizes due to fluid flow. The elution volumes of suspended particles or polymer molecules are derived for various tube geometries. Following Taylor, the effects of diffusional broadening of the volume elution peak for finite size particles are discussed and a criteria for separation is given. It is found that particles very similar in size can always be separated. A scheme for separation by flow on a continuous basis is proposed.

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 application of this procedure results in an approximate expression for the combinatorial term which allows one to predict qualitatively the change in the entropy of packing as a function of stretch. It is found that the entropy of packing has the proper behavior to explain the initial deviation of the experimental stress-strain curve from the previous theoretical predictions.

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 matrix of transition probabilities. Characteristic polymer dimensions near a surface are found to be larger than they would be away from the surface. The diffusion equation is then generalized so that it describes a polymer chain with self-excluded volume. Energy effects are also discussed.

~n~ral formula;> for .the then;nodynamic P!operties of amorphous polymer phases are obtained from statlstlc~l mechamcs, with the aid of the lattice model, in a manner which avoids the use of restrictive assumptwns 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-Hug~ins 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 inf?r~ation m.ay be obtained by application of the general formulas to models descriptive of. the mol~cular chal~s In quest~on. As an example, the results of Flory for semifiexible linear chains whose stJf'fI:ess anses excluslve~y from In.tramolecula.r nearest neighbors are obtained as a special case. (The conventlOnal thermodynamiC properties of polydlsperse systems of chains of this type are shown to depend on the number average molecular weight.)