Polymer entanglement > How entangled is a polymer melt?

A fundamental property of melts of long flexible polymers is that they are entangled, like cooked spaghetti in a bowl. It is relatively easy to pull one molecule (or strand of pasta) along its own path, but difficult to move it "sideways", because of other molecules running in other directions that cannot be passed through.

If we watch a movie (courtesy of molecular dynamics simulations) of a polymer melt, we can see that a molecule on short timescales tends to wiggle about in what appears to be a "tube", constraining its "sideways" motion. The tube diameter is a "material parameter" — depends on the kind of polymer, but not the length of the molecules. A useful correlation exists between the tube diameter — which is a dynamical property, hard to measure but important for how chains move — and the stiffness and bulkiness of the polymer chains, which are simple geometric parameters, easy to measure. This correlation can be used to predict the stiffness of rubbers and the elasticity of flowing polymer melts and solutions.

But nowhere in that correlation was any mention made of uncrossability, which was the reason for there being a "tube" in the first place! Which begs the question, how can we relate the tube diameter to uncrossability of the chains?

In recent work, we have provided an answer to this longstanding question, using some ideas and results from the mathematical theory of knots. Recent work by topologists goes a long way towards being able to "tell one knot from another", by making calculations based on the "crossing diagram" — the pattern of which strand crosses over which strand where, when the knot is mashed down onto a sheet.

In extensive simulations, we have generated many millions of topologically distinct configurations of a single long ring polymer, at melt density, in 2d periodic boundary conditions, entangled with its periodic images. We have written a program to efficiently compute the HOMFLY characteristic polynomial to (nearly) uniquely identify knots, with its definition extended to identify the 2d periodic "chain-mail patterns" created by our ring polymer entangled with its images.

In this way, we can compute the topological entropy of our one-ring melt, defined in the usual information-theoretic fashion as ST = -Σpk log pk where pk is the probability that the ring is found in the kth distinct knotted configuration. We expect this entropy to vary with the ring length ultimately as N/Ne. That is, a chain in a melt entangled with its surroundings makes essentially one "decision" every entanglement strand, of which way to go. Thus measuring ST allows us to determine Ne from purely topological considerations — and we find a value in very good agreement with more pedestrian ways of determining Ne.

We can also relate topology to entanglement of our one-ring melt by a simpler quantity, which is the probability that our ring is not actually knotted. This "unknot probability" is unity for short rings and approaches zero for long rings, crossing over at a characteristic length proportional to Ne. This approach to determining Ne by topology is nice because it is much simpler to detect an unknot than to uniquely identify a knot.

We can use these tools to explore how far a given chain can wiggle before it typically cuts through another chain in the melt, and thereby "changes the knot" made by all the chains in the system. How much wiggle room a chain has depends on how close other chains are (which depends on chain stiffness and bulkiness; bulky chains stay far apart, stiff skinny chains can pass close by). In this way we can connect topology, geometry, and tube diameter.

(For more information: J. Qin and S. T. Milner, Counting polymer knots to find the entanglement length. Soft Matter **7**, 10676–10693 (2011).)

Polymer entanglement > How can we "see" the confining tube in entangled polymer melts and solutions?

Modern theories of entangled polymer dynamics rely on the concept of a "tube", which replaces the uncrossability constraints on a given chain arising from nearby chains, with a confining "tube potential". The centerline of the tube is called the primitive path.

The tube diameter a, different for each kind of polymer, sets the scale over which a chain can wander transversely from the primitive path before nearby uncrossable chains impede its motion.
The entanglement length *Ne* is the arclength of a chain strand of dimension a. The tube diameter determines how "stiff" a polymer melt is, in responding elastically to a sudden strain.
The smaller a is, the more entangled and stiff is the melt.

The tube diameter together with the monomeric friction factor are the two "microscopic" parameters in the highly successful modern theory of flow behavior of entangled polymer melts. As a dynamical construct, the tube is difficult to "image" directly in experiments. Thus attempts have been made to “see” the tube in simulations.

Prior to our work, the main approach to visualize the tube in simulations has been various "chain-shrinking" methods,
in which the free ends of chains in the system are fixed, and the tension along the chains effectively increased by some means,
until the configuration of each chain becomes a set of straight segments between successive pairwise contacts with other chains.
The primitive path is approximated as the stepwise linear path, and *Ne* as the number of monomers between contacts.

Although chain-shrinking methods have been helpful in visualizing the tube, they are limited because of the "damage" done to the chain conformations by the shrinking process. We would prefer a "nondestructive" method, in which the tube can be observed simply by watching the chains fluctuate, with no external forces applied.

To find the tube without touching the chains, we turn to an earlier approach, of simply averaging the chain trajectory over a short time *Ta*,
long enough for an entanglement strand to explore the tube but short compared to timescales for stress relaxation (Rouse and reptation times).
The time-averaged chain path approximates the primitive path.

We improve on this method with *isoconfigurational averaging*: from the same starting configuration, we make many short movies of the subsequent chain motion,
each with different random initial velocities. Thus we average not only over where the monomers of a given chain did wander in one short movie,
but a cloud of points for where each monomer traveled in a host of different movies.

Isoconfigurational averaging gives greatly improved statistics for finding the primitive path with a relatively short time average.
The resulting primitive path is a smooth curve, with a persistence length of about *Ne/2*. Because we do not shrink the chains,
we can determine the tube confining potential (well described as harmonic), quantify how the tube diameter varies along a given chain,
and measure how the tube broadens near a free surface (where absent entanglements from the "vacuum side" reduce chain confinement).

To carry out our studies of tube properties, we prefer to simulate entangled long *ring* polymers.
The rings serve as a proxy for a melt of long entangled chains, with the advantage that there are no free ends,
at which the chain relaxes quickly and the tube is ill-defined. With rings, the state of entanglement is permanent,
which is ideal for watching chains fluctuate in their tubes.

Of course, we must first "topologically equilibrate" the melt of rings, which we do by allowing the chains to cross occasionally.
We verify equilibrium by watching the ring conformations fluctuate, then turn off chain crossing to observe the tube. Finally,
it is both desirable and sufficient to simulation only one or a few rings, each many *Ne* long, at melt density in periodic boundary conditions,
entangled with their periodic images. This helps to mitigate whatever effects of finite ring length that would arise with multiple shorter rings.

We are currently using our isoconfigurational averaging method to explore the effects on the tube of 1) deformations of the melt (uniaxial extension and compression), and 2) tension applied to linear chains. At issue is the fundamental question, relevant to modern theories of nonlinear rheology of entangled polymers: what are the properties of the tube for chains that have been stretched or aligned by flow?

(For more information: W. Bisbee, J. Qin, and S. T. Milner, Finding the tube with isoconfigurational averaging. Macromolecules **44**, 8972–8980 (2011).)