Dynamics of chains grafted on solid wall during polymer melt extrusion

The objective of the present work is the mathematical modeling of the dynamics of polymer molecules grafted on a solid boundary during polymer melt extrusion. This topic is closely related to the long-standing problem of polymer flow instabilities encountered in industry when extruding melts. In order to describe the behavior of the tethered chains, we introduce the bond vector probability distribution function (BVPDF) which appears to be a simple, yet effective mathematical 'tool'. The bond vector, i.e. the tangent vector to a polymer chain depending on the position along the chain and on time, describes the local geometry via its direction and the local stretching of the chain via its length. The BVPDF contains all information about the geometry of the ensemble of chains. Via averaging over the BVPDF we can calculate all interesting macrsocopic quantities, e.g. the thickness of and stress in the layer of tethered molecules. The time dependence of the BVPDF yields the time evolution of the system. We derive the equation of motion for the BVPDF taking into account all important mechanisms, such as reptation and (convec-tive) constraint release. Besides that, we show that all macroscopic quantities of practical interest can be expressed via second order moments of this distribution function. Prologue Previous studies of polymer melt flow near solid boundaries have shown that there are two regions with different rheological behavior. The first one is far from the solid wall and then is referred to as bulk flow area. It is only occupied by bulk molecules (chains that do not contact with the wall). The second region is close to the wall and is " populated " by both bulk and tethered molecules. It will be referred to as interfacial layer throughout the text. The goal of the present paper is to compute the thickness of the interfacial layer for different flow regimes. This study is a step toward complete understanding of what is really going on near the solid wall. The issue concerned is closely related to the problem of flow instabilities encountered in polymer extrusion. We begin with the simple case of slow flows and low grafting densities. In this situation, the thickness h can be estimated from the freely-jointed chain model. We conjecture that it should be of the order of the mean end-to-end distance and hence h ≈< (R(L, t) − R(0, t)) 2 > 1/2 = √ N b (1) …