Marc L. Mansfield

Solution of the growth equations of a sector of a polymer crystal including consideration of the changing size of the crystal

Abstract We obtain a solution for the nucleation-controlled growth of a sector of a polymer single crystal that treats directly the changing size of the crystal. The problem is treated by solving a pair of differential equations with moving boundary conditions. A steady-state solution is obtained for the case in which the two boundaries move apart at the same rate. The steady-state solution is inherently regime II for all possible values of the model parameters. The growth-front profile is a section of an ellipse.

Surface adsorption of model dendrimers

Abstract Lattice model dendrimers interacting with an adsorbing planar surface are studied by computer simulation, where G is the number of generations of the dendrimer and A is the interaction strength. With increasing A , dendrimers are observed to spread out and flatten down on the surface, as expected. In certain regions of G - A space, two competing configurations states, S 2 and S 3 , are observed. In S 3 all three dendrons are adsorbed on the plane. In S 2 two dendrons are adsorbed, while the third sits up and away from the surface. G - A space divides into five separate regions: a desorption region where A is too weak to maintain adsorption, a weak adsorption region in which the dendrimer tends to maintain contact but with only weak perturbation of its shape, a region in which S 2 and S 3 compete because they have comparable free energies and are separated by a modest free energy barrier, a region in which S 3 dominates because it has lower free energy, and finally a region in which S 2 and S 3 both have high stabilities because they are separated by a large free energy barrier.

The random parking of spheres on spheres

Given a ‘‘target’’ sphere of radius r1 and ‘‘probe’’ spheres of radius r2, we consider, as a function of r2/r1, how many probe spheres, on average, can be attached to the target sphere if (1) the attachment sites are chosen at random, (2) the probe spheres are not permitted to overlap, and (3) each attachment is irreversible. We also consider two separate extremes for selecting new attachment sites: Each probe sphere is either permitted to diffuse into place from a large distance, or the attachment site is chosen completely at random. Diffusion‐controlled attachment produces a slightly higher packing density than completely random attachment.

Dendron segregation in model dendrimers

Abstract Monte Carlo calculations on model dendrimers demonstrate that individual branches (‘dendrons’) of an isolated dendrimer spontaneously demix and are well segregated at equilibrium even when they are chemically identical. This is a result of the particular architecture of this class of macromolecules. It is conjectured that this segregation effect will disappear under high density conditions, e.g. in poor solvents, or at the highest attainable generation number, or in neat fluids of dendrimers. The dendrimers possess fractal self-similarity over a rather narrow scale of lengths with fractal dimensions of the order of 2.4 to 2.8.

The coil–stretch transition of polymers in external fields

We consider the exact statistical mechanical properties of a simple cubic lattice chain with one end anchored at the origin and with all other segments experiencing an external potential of one of two forms: −c‖x‖α (even Hamiltonian) or −c sgn(x)‖x‖α (odd Hamiltonian), for α an arbitrary exponent greater than zero and for c an arbitrary field strength. The problem is exactly soluble numerically for N not too large and for arbitrary α by transfer matrix techniques. In addition, the odd Hamiltonian with α=1 is especially simple to solve, yielding closed form expressions for a number of properties. Both Hamiltonians exhibit a first‐order phase transition at c=0 in the limit of large N. The even Hamiltonian exhibits a coil (c 0) transition. The odd Hamiltonian exhibits a left‐stretched (c 0) transition. For N sufficiently large and for α>1, the entire chain participates in the transition, becoming completely stretched for c only slightly greater than zero. When α<1, t...

Model of the glass transition

This paper examines the three well‐known empiricisms that accompany the glass transition of fragile glass‐forming liquids: the unusual thermodynamic behavior of supercooled liquids that is usually called the Kauzmann paradox, the temperature dependence of the relaxation time as embodied in the Vogel–Fulcher law, and the time‐domain relaxation law usually known as the Kohlrausch–Williams–Watts relaxation function. Mutually consistent descriptions of all three phenomena are presented. The Kauzmann paradox can be explained by recognizing that the ground states of both the liquid and the crystal, and also the dominant excitations of these two ground states, are similar in many ways. This implies that the two phases have similar thermodynamics at low T. The Vogel–Fulcher law results from the assumption that localized regions of the liquid must be excited above a certain threshold enthalpy before they can relax; as the temperature falls, this threshold enthalpy becomes less accessible and the apparent activatio...

volume relaxation of glasses

The rate of volume relaxation of glasses is computed by assuming that the volume changes only because of the diffusion of free volume in and out of the sample through the surface. The relaxation curves are strongly affected by the assumed diffusion law, which means that such experiments provide a potentially powerful tool for determining the process by which free volume diffuses in glasses. For ordinary, i.e., Fick’s law, diffusion, the relaxation curves have exponential tails, while for anomalous diffusion of the sort introduced by Shlesinger and Montroll, the curves are predicted to have inverse‐power law tails. Comparison with experimental results on glassy polymers permits us to rule out Fick’s law diffusion completely, and strongly supports the anomalous diffusion law of Shlesinger and Montroll.

Lattice models of semicrystalline polymers: the Flory theorem implies that mean-field predictions are exact

Abstract The statistical properties of several different ensembles of properly packed (i.e. non-mean-field) lattice chain systems that model the amorphous domains of semicrystalline polymers are studied by Monte Carlo calculations. The models are shown to obey accurately mean-field predictions as long as comparisons are limited to isotropic regions. This is seen as a direct consequence of the Flory theorem. In fact, this calculation is a particularly sensitive computer simulation test of the Flory theorem, because it permits calculations to be performed in the absence of chain ends.

Statistics of polymer chains near a notch and consequences for twin polymer single crystals

Abstract It has been argued that flaws exist in the standard nucleation theory of polymer crystal growth because this theory seems to predict that the notches observed in twin crystals of polyethylene should fill out so rapidly as not to be observable. However, I argue that the observed growth in the notch is reconcilable with standard nucleation theory. When proper consideration is taken for the ability of polymer chains to diffuse into the notch, the observed nucleation rates are not surprising. I consider four separate cases. In the first case, I assume that polymers, although attracted by the growth front, are not attracted strongly enough to be adsorbed prior to crystallization. In the second case, I assume that polymers adsorb readily on the crystal face, and then crystallize very near the point of first contact. In the third case, I assume that chains adsorb on the growth front prior to crystallization and that they are able to diffuse about on the growth front through large distances prior to crystallization. In the fourth case, polymers adsorb rather weakly, and reversibly, attaching and reattaching a number of times prior to crystallization. I am able to rule out the second case, while all the others predict a lower than expected nucleation rate and also predict the observed molecular weight and temperature trends.

A generalized Eden model of polymer crystallization

Abstract A generalization of the Eden model is proposed as a model of polymer crystallization. The model exhibits all three growth regimes. It produces the Seto-Frank model in one limit and demonstrates departures from the Seto-Frank model induced by discreteness of the crystal. It is able to predict curved growth fronts; in fact it predicts that circular crystals are to be expected in regime III.