Entanglements do exist, despite what atheists say

By entanglement I mean some discreet event between two or more chains. Discreet means that I can say at what time the entanglement appeared, when did it disappear, which monomers of which chains have participated etc. This might be contrasted with the tube or mean field approach which assumes a more uniform and continuous entanglement field, created by many chains.

In order to see entanglement clearly in molecular dynamics (MD) simulations, we have to get rid of fast and rather chaotic motion on short time scales (below tau_e). One way to do this is by time and ensemble averaging. Here is the video of two chains selected from the melt, which are clearly entangled. The details of the simulations can be found in our paper, the chains are slightly semi-flexible (kb=3, N=100). What you see is not the chain position, but position of tube axis of each chain. The tube axis for each configuration is defined in the following way: I start m jobs (m=50 in this case) from the same initial configuration, run them for about tau_e, average position of each monomer over this time, and then average results over ensemble. The resulting tube axis is much much smoother then the chains themselves, and (hopefully) most of fast degrees of freedom are averaged out. This creates some very clear pictures - enjoy.


The green chain has its ends fixed, the purple is free. In the beginning the entanglement is very strong. At frame 19 it becomes weaker (look at the chain end), and at frame 53 they completely disentangle.

The disentanglement process is clearly illustrated by plotting the minimal distance between the tube axis of this pair of chains as a function of time. If the chains are entangled, it remains very small (of order one atom).

Surprisingly, this is not an exotic case. In this particular melt (with about 15-20 entanglements per chain according to usual definitions), almost each chain has such a strong or "double" entanglement. But of course they are not all the story - there are much more weaker entanglements.

The reasons for starting this blog

This blog was inspired by reading a book called Wikinomics - a very optimistic and up-bit praise of openness and sharing. One of the main ideas of the book is that these two do not necessarily contradict competition, even in business. For example, IBM spends now $100M a year to fund Linux development, and find that it saves money. Another example is human genome project, partially sponsored by big pharmaceutical companies, but all results are published in public domain. The companies benefit from sharing if several conditions are in place:

• they encourage a wide community to co-create new intellectual property (IP). This widens the number of enthusiastic consumers (who become prosumers using book's terminology), and also helps to advance the field as a whole.
  
• the shared part of IP is not their main area of profit-making. For example, IBM makes money on servers and network solutions, and they benefit from having a free stable operating system.
• Putting results in public domain prevents competitors from patenting the information crucial to the business.
  

If it works is business, can it also work in science, and, in particular, in our beloved field of entangled polymer dynamics? At the moment, we hold on to our ideas until they are published or presented at the conference. Some people do not present unpublished results at conferences, but majority do. Most of people are afraid that somebody will publish their ideas before they do, or will develop them further and better than they do. So can we really benefit from sharing and more openness?

This blog is an experiment designed to answer this question.

However, of course I have a bit of theory as well. This are the arguments why me and you might benefit from contributing to these pages and share your ideas with the community:

• you might have more ideas than you and your co-workers can test. In this case you can post some of them here. I can envision several possible positive outputs
1. somebody tried it already and rejected
2. somebody wants to try it and collaborate with you
   
3. somebody can criticize this idea in the comment and explain why it will not work


• you want to discuss a subject you feel passionate about and want to hear the comments from your colleagues across the world. You can do it here at leisurely pace rather than wait until the next conference and discuss it in a queue for coffee with each individual separately.
  
• You want to explain your ideas presented elsewhere in this informal forum, and to hear the feedback.

Of course I will not be able to judge whether the experiment agrees with the theory since I am a bit biased - I want it to succeed because I invested some time in it. But the potential benefits will be much bigger than that. The best possible result will be that we as a field advance faster and more effectively than other fields, and will be viewed from outside as the successful branch of science worth investing into. We might even solve the whole problem in our lifetime.

Problem formulation

It is common to start a proposal or a new paper in our field saying that for the last 40 years entanglements were treated in the framework of the tube theory. This is both good news and bad news for the tube theory: 1) it's good because in 40 years nobody came with a better alternative; 2) it's quite embarrassing that the tube theory was not able to solve the problem of entanglements for 40 years. The second point is of course my personal view, which is discussed in details in this preprint for JNNFM. Many people would disagree and indeed told me that most problems in my area are more or less solved and one should move to more exiting areas of physics. But there are others who think that we did not even scratched the surface of the problems since we do not even know what entanglement is, or how to define the tube. As you probably guessed, I am inclined to agree with the second point of view (perhaps that is why I ended up in Mathematics Department), but that does not mean that I think the tube theory is wrong. There is too much experimental evidence which shows that something like a tube or reptation exists. But this is as much as one can say without properly defining what the tube or entanglement actually is.

I strongly believe that if we want to solve the problem of entangled polymer dynamics, we must start at the beginning, or at the bottom of the length-scales ladder. One has to define an entanglement or a tube. But in this post I do not want to discuss how to do it - this is definitely a major challenge, and groups around the world including ours are working on it. What I want to discuss is the following: how will we know that we found it? Seriously, can we unambiguously check that the entanglement or a tube someone found is indeed everything we need to know about constraints other chains impose on a particular chain? I think the answer might depend on whether you are talking about entanglements or tubes, so I will consider these two cases separately.

I think the main difference between entanglements and tubes is that entanglements are discrete events, whereas the tube is continuous. Consequently if I operate within entanglement framework, I should be able to distinguish whether a particular chain has 6 or 7 entanglements, and the main parameter of such theory would be Z – number of entanglements per chain. The tube model assumes existence of a primitive path or a tube axis, and that the field from other chains restricts the motion of a particular chain to be somehow within a tube diameter from the primitive path. Thus, the main parameter of the tube model is tube diameter and/or tube persistence length. In this sense, tube model is a bit more mean field than entanglement model, since it discusses only average deviations from the primitive path. But of course nothing stops us from discussing fluctuations of the tube diameter or tube persistence length.

Entanglements

Assume someone gives you an algorithm which finds all entanglements in your system. I will assume that each entanglement is a list of monomers and chains which participate in it, and possibly some additional information (like the strength of entanglement or something like this). In case of binary entanglements this is just saying that the monomer i of chain A is entangled with monomer j of chain B. For example, primitive path analysis (PPA) provides such information. How can I be sure that this is indeed all I need to know, how can I check that this information is enough to model dynamics of my melt?

One straightforward answer would be to use NAPLES model of Yuichi Masubuchi. Do molecular dynamics of entangled polymers, perform PPA on the trajectory, feed this information to NAPLES, and compare all observables. Somebody should do it, all components and codes are available. We did something similar with Sathish Sukumaran comparing slip-springs model with MD, but we fitted slip-springs parameters rather than extracting them from MD (submitted to Macromolecules). There are however significant drawbacks in such procedure. The disagreement might have two principally different reasons: either we did not find all entanglements, or NAPLES model assumed wrong dynamics for correct variables. In turn an agreement might be a consequence of cancellation of errors or a result of insensitivity of particular set of observables to the structure of entanglement network.

Can one formulate a better test, which does not rely on a particular model? I am thinking along the following lines. Entanglements are supposed to be a good set of slow variables. That means that they should be enough to predict evolution of the system on long time scales. That in turn means that if I tell you all entanglements on my particular chain, you should be able to tell me the stress, structure factor and other similar static observables carried by this chain. But this stress should be the stress averaged over fast fluctuations somehow, for example by averaging over time about tau_e. Any ideas?

Tubes.

What is the primitive path or a tube axis? What we know is that if we zoom out to the length scales much larger than tube diameter, the chain motion will look like one-dimensional motion along this primitive path. The definition accurate on the length-scale of tube diameter is problematic: it can be the minimal distance connecting two ends, or a tube axis, i.e. the axis of maximum monomer density, or something like a mean path. But suppose again someone gives you all the primitive paths in your system (plus maybe a tube diameter). How do you know this is all you need?

Do not you think it would be really nice to have some clear answers to these questions? In mathematics the situation is clearly like this: there are many open problems posed, but when one is solved, everybody (more or less) knows that this is indeed solved and one can go on and play with other problems. What a wonderful life!

New definition of the tube diameter

Suppose you know the trajectory of your chains (say from molecular dynamics or single chain stochastic models such as slip-links or slip-springs). How can you calculate the tube diameter?

Here is one possibility: I created a new correlator which calculates the following correlation function:
where r_{ik}(t) is the position of the monomer i of chain k at time t, Nc is number of chains, N is number of monomers in each chain, and T is the total time of your trajectory. In other words, I calculate the average distance from one monomer at particular time to the monomer which was closest to this position some time ago. To start with, I will consider this quantity for middle monomer only, i.e. i=N/2.

If your chain only reptates (for example slithering snake algorithm), this quantity will be zero until time of order reptation time. In reality it will of course be finite, and if it is constant for some time, I suggest this constant will be a good measure of perpendicular thickness of the tube, or tube diameter. This should not be confused with the tube persistence length, which is what Doi and Edwards call "tube diameter" (in their theory these two quantities are assumed to be equal, whereas they can be very different, for example in semiflexible chains).

Here are few results:

This figure shows results from MD simulations of the most entangled chains I can simulate (kb=3, N=200, Nc=400, see our paper for details). The line shows a(t) for the middle monomer, whereas points show (from bottom to top): center of mass dissplacement g3(t), middle monomer displacement g1(N/2,t) and end monomer displacement g1(0,t). One can see that a(t) is almost constant (about 4-5), but not quite - of course I blame constraint release for that. To illustrate this, I calculated the same thing from the slip-springs model (N=48, no CR, Ns=0.5, Ne=4)
The plateau is now very clear at about a^2=0.72.

More investigation is needed, including MD with fixed ends. I would also like to create analogous correlator in parallel to the tube direction - hopefully in the next posts.