Markoff Chain Model for Growth of Polymer Single Crystals

Abstract

A theory is presented wherein the development of a polymer single crystal is envisioned as the successive coherent addition of layers of segments of polymer molecules, each layer acting as the substrate for the succeeding layer. The substrate is assumed smooth and the distribution of segment lengths in the next layer is assumed to be controlled by equilibrium considerations. The theory predicts that as the supercooling is increased the energy gained due to the bulk energy of crystallization tends to overwhelm the energy lost due to formation of new surfaces. This has the result of making the average segment length of the adding layer closer and closer to the substrate length. In some cases it even exceeds the substrate length. The net result is that the temperature dependence of ultimate crystal thickness is not as great as was implied by the (ΔT)—1 laws of previous theories. The theory has been satisfactorily fitted to the experimentally observed dependence on temperature of single crystals of linear polyethylene grown from dilute xylene solution. The parameters used to fit theory to experiment are a lateral surface energy of 13.3 ergs/cm2, an end (fold surface) energy of 110 ergs/cm2, and a solution temperature of 113°C. The theory also indicates that the surface of the polyethylene crystals is remarkably smooth with an average roughness of only one to two angstroms. Predictions are also made concerning the profile of a single crystal in the region of its primary nucleus (central region). It is shown that hollows, sharp spikes, or flat topped mesas of considerable extent may be observed depending on the supercooling.