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.

A melt or solution of long entangled polymer chains flows like honey — very viscous, and even “elastic”, for example snapping back when you cut a flowing stream with scissors.  This behavior comes from the stretchy, springy nature of the long molecules, and the fact that they are entangled together.  Different kinds of polymers — made from different chemical “beads” strung together in a long necklace — are more or less elastic in this way, depending on how entangled they are.

Isoconfigurational averaging 

“How can we “see” the confining tube in entangled polymer melts?”

To study entanglement with computer simulation, we made “movies” of how the molecules wiggle about in an entangled melt.  The molecules try to move sideways, but run into neighboring chains oriented crossways, and bounce back.  The molecule acts as if it is trapped in a soft “tube”.  The more entangled it is, the wider the tube.  We can “see” the tube in simulations, by averaging over all the ways the chain can wiggle from the same starting arrangement.  The set of places the chain can wiggle to, appears as a “cloud” of points around an average entangled path.  The width of the cloud reveals the diameter of the tube.

The “cloud” of points the monomer “beads” of a chain can visit, by wiggling inside confining “tube”.

It turns out that stiff, skinny polymer chains are more entangled — they can get closer to each other, and get in each other’s way more effectively — while flexible, bulky chains take up more room around themselves, which keeps the entangling effects of other chains further away.

Thus the tube diameter is a “material parameter” — it depends on the kind of polymer, but not the length of the molecules.  In fact, 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.

Scaling relation between entanglement volume and “packing length”, a measure of chain bulkiness and flexibility, for a wide range of polymer structures.

Bisbee, W., Qin, J., and Milner, S. T. “Finding the Tube with Isoconfigurational Averaging” Macromolecules 44, no. 22 (2011): 8972–8980. doi:10.1021/ma2012333

Qin, J., So, J., and Milner, S. T. “Tube Diameter of Stretched and Compressed Permanently Entangled Polymers ” Macromolecules 45, (2012): 9816–9822.

Qin, J. and Milner, S. T. “Tube Diameter of Oriented and Stretched Polymer Melts” Macromolecules 46, no. 4 (2013): 1659–1672. doi:10.1021/ma302095k

Cao, J., Qin, J., and Milner, S. T. “Simulating Constraint Release by Watching a Ring Cross Itself” Macromolecules 47, no. 7 (2014): 2479–2486. doi:10.1021/ma500325z

Relating entanglement to knot theory

“How can we connect the tube to uncrossability?”

Nowhere in the correlation between entanglement volume and packing length, is any mention made of uncrossability — which was the reason for there being a “tube” in the first place.  So how can we relate the tube diameter directly to uncrossability of the chains?  

To answer that question, we use some powerful ideas 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 table top.

Knot theorist’s “periodic table”, of distinct irreducible knots in order of complexity.

Using knot theory, we measure how entangled is a polymer melt by counting how many different knots it can tie, if allowed to explore all possible arrangements.  We developed an efficient computer algorithm using knot theory tools to count how many of each kind of knot a small polymer melt could randomly tie.  The more knots a ring can tie, the more entangled it is.  In fact, we find that the topological entropy is proportional to the entanglement length. 

In a similar way, a ring of length greater than the entanglement length Ne is increasingly likely to be knotted. We count the number of “unknots” (a much easier task) for single-ring melts as a function of length, and determine Ne. With this method, we have explored the effect on Ne of chain stiffness and solvent dilution, and find variations consistent with experiment. 

Unknot probability p0(N) versus length N: rings with N > Ne become knotted. Ne is smaller for
stiff chains (left curves), larger if diluted by solvent (right curves).

Qin, J. and Milner, S. T. “Tubes, Topology, and Polymer Entanglement” Macromolecules 47, no. 17 (2014): 6077–6085. doi:10.1021/ma500755p

Qin, J. and Milner, S. T. “Counting Polymer Knots to Find the Entanglement Length” Soft Matter 7, no. 22 (2011): 10676–10693. doi:10.1039/c1sm05972f

Dynamical regimes in entangled ring melts

“What can we learn about how entangled polymers move, by watching a melt of entangled rings?”

Entangled linear chains in a melt can reptate — diffuse back and forth inside their own tubes, exploring new configurations, without crossing other chains.  On shorter timescales, the chains wiggle about in their tubes locally, moving like a Slinky toy, stretching in one place and shrinking in another, under the action of thermal agitation.  

At first glance, motion of a melt of entangled rings seems very different; the rings can’t escape their tubes and diffuse away, as linear chains do.  But actually, all of the regimes of motion that linear chains undergo in their tubes, rings exhibit as well, except for the final escape.  So we can learn a lot about entangled linear chain motion by watching entangled rings.  

Bead MSD regimes. I: Beads move by Rouse dynamics locally within the tube. II: Beads move by Rouse dynamics along the tube. III: Beads reptate around the ring. IV: Beads traverse the tube many times; MSD reaches a plateau.

An important advantage of rings is:  a ring has no ends, so all its monomer “beads” are structurally equivalent.  This means ring motion is simpler theoretically, because there are no overall contour-length fluctuations or constraint release events, as for linear chains.  And, because all monomers are equivalent, we can average over the motion of all the beads to get better statistics.

Using this idea, we have simulated a self-entangled melt of a single long bead-spring ring polymer, to carry out a precise test of Doi-Edwards theory for entangled polymer dynamics.  Entangled melts have four regimes of dynamics, as reflected in the monomer mean-square displacement (MSD) versus time.  The first three regimes are the same for entangled linear chains and rings:  local Rouse (Slinky) motion in a part of the tube, overall “collective” Rouse motion in the tube, and reptation in the tube.  (The fourth regime for linear chains is free diffusion, which for rings is suppressed by the permanent entanglements.)  

Indeed, we observe all four dynamical regimes in a family of self-entangled rings of different length.  Furthermore, we can write a relatively simple theory for the MSD versus time, which agrees very well with simulation results — thus providing a strong test of the Doi-Edwards theory.

Bead mean-square displacements (MSD), for three rings with N = 400, 800, and 1600; MD results (solid) and tube theory predictions (dashed).

Qin, J. and Milner, S. T. “Tube Dynamics Works for Randomly Entangled Rings” Physical Review Letters 116, no. 6 (2016): doi:10.1103/PhysRevLett.116.068307