Chapter 3 Statistical Properties of Constrained Lattice Trees
4.1 Entangled Loop Strands
4.1.1 Ideal Chains
Based on the analysis in Section 3.3, a simpler model of entangled ring polymers can be made by adopting the ansatz that the ring polymercompactifiesin such a way as to form a duplex structure in which each tube segment [Cates and Deutsch, 1986] contains an outgoing and a returning segment of the ring polymer; see Figure 4.1. The effective length of the tube is reduced by a factor of two if assumed ideal [Zimm and Stockmayer, 1949], but the chain is now guaranteed to satisfy its topological constraints and remains unknotted.
This represents an explicit set of microscopic configurations that can be shown to satisfy all the topological constraints rigorously. This set may fail to include
Figure 4.1: A schematic diagram showing three unbranched, compactified ring poly- mers (solid curves) and the tubes (dash-dotted curves) provided by their entangle- ments with a background gel and/or neighbouring rings (not shown). The labels identify the nature of penetrations on Polymer 1.
some equilibrium chain states in the melt, where the duplex structure may permit penetration between the duplex pair (violating compactification) and be branched. However, it is the primary aim of this research to study the effect of such pene- trations perturbatively (where they remain rare). While one can also suppress the branching by setting the mesh size of the background gel or neighbouring rings to be much smaller than the persistence length of the chain, it is anyway natural to study the unbranched limit first. It is likely that the onset of the topological glass transition that we describe below will be shifted if we permit branched structures, e.g., removing the background gel or choosing one with a large mesh size. However, it is difficult to see how itsexistence could be affected by branching. The novelty of our approach lies in the way in which we account for penetration events; see Figure 4.1. These do not violate topological constraints and can be thought of as a perturbative relaxation of the compact chain ansatz introduced above.
In Figure 4.1, the labels identify the nature of penetrations on Polymer 1. Here Polymer 2 has actively penetrated Polymer 1, creating an associated passive penetration in the corresponding tube segment of Polymer 1. Polymer 3 has been penetrated by Polymer 1, resulting in a passive penetration to Polymer 3 and an active penetration to Polymer 1. The passive penetration on Polymer 1 will remain until one end of Polymer 2 has diffused through that tube segment. Until that
happens the motion of Polymer 1 is restricted by this penetration, which prevents either end diffusing through this tube segment. The active penetration on Polymer 1 will be lost as soon as either of its ends moves through the tube segment containing this penetration, simultaneously annihilating the corresponding passive penetration on Polymer 3.
4.1.2 Fluctuation, Length Defects and Reptation
Before discussing the detailed calculation of the penetrating loop strands, it is in- structive to review the most successful mean-field model of entangled linear poly- mers. As proposed by de Gennes [1971], the fundamental relaxation mode of an entangled linear polymer is believed to be the propagation ofkink gas, a series of non-interacting length defects, along the polymer chain; see Figure 4.2. This is based on the concept that the non-crossing constraints imposed by the surrounding chains prevent a chain laterally moving through them. The length defects hence tend to move randomly along a curvilinear path, which is correlated to the tube-like confinement of the surrounding chains. The random motion of these length defects, of course, originates from the thermal fluctuation of the length and the conformation of the polymer chain itself.
Later Edwards refined this concept and built a detailed analytic model (now often referred to the Doi-Edwards tube model) to describe such processes. The most important result of this model is the disengagement time,
τd(0)= ζ0N
3b4
π2k BT a2
, (4.1)
where ζ0 is the friction constant of a monomer, N the degree of polymerisation, b
the bond length andathe tube width. This is the time needed for a primitive chain, the statistical representation of a confined polymer, to disengage from the original tube it was confined to att= 0. Notice that the parameterais of the order of the mesh size of surrounding chains, which is approximately the entanglement lengthle
[Doi and Edwards, 1986]. In later discussions this disengagement time is rescaled byζ0 →ζ,N →Ns andN b2/a→Lc, whereζ is the friction constant of a segment,
Ns the number of segments and Lc the contour length of a primitive chain. The
rescaled disengagement time can be seen as a special case without penetration of primitive chains involved, therefore it is denoted asτd(0) to stress the fact that there is no penetration.
It is noticeable that, as shown in Figure 4.1, the ring polymers have no chain end to relax length defects as that in the Doi-Edwards tube model. This is one of
(a)
(b)
(c)
Figure 4.2: The tube model describes the relaxation process of entangled linear polymers as follows. Each polymer can only move along a curvilinear path—the primitive path (dashed lines)—due to the constraints of neighbouring chains. The kinks, also called the length defects, randomly diffuse along the path until them move to the ends. The fluctuation at the ends also creates new kinks simultaneously. The chain ends essentially explore new paths toward random directions, see (a)–(c), and eventually the whole chain disengages from the original primitive path. In this figure the tube is not shown.
the reasons why many see developing analytic models for entangled ring polymers as a challenge in polymer physics. Without the chain ends, the most accessible mean-field theory seems to have no way to describe the dynamics of ring polymers. However, in certain conditions, the fluctuation of the length and the conformation should still provide a similar relaxation mode to that of reptation. Combined with the discussion in Chapter 3, it is possible for highly entangled ring polymers to be modelled as linear objects with relaxation via a form of reptation. In this chapter, we will demonstrate a simple model of penetrating loop strands which is built on the basis of the Doi-Edwards tube model.