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!
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!
