So, I've been looking into polymer chemistry to model breaking polymers for a biological model, and I've been a bit stumped by the concept of persistence length.

I know that if a polymer is significantly longer than its persistence length, it needs to be treated as though the ends are completely independent of each other thanks to thermal fluctuations bending it all over (the analogy I have is treating the polymer like a piece of cooked spaghetti). If I model a polymer like this, I need to use the words "3 dimensional random walk" to describe its shape mathematically.

I also know that if a polymer is significantly shorter than its persistence length, it can be treated like a rigid rod that might be a bit bendy, like a steel rod. This is significantly easier to model: I can just treat it as a line/arc that moves through space as a single particle.

But what happens if a polymer's length is about equal to its persistence length? How can the polymer be modeled then? What precisely does it mean for the polymer's movement?

EDIT: I've been debating the merits of just treating it like a flexible rod (like the significantly shorter than persistence length example) and calling it good just because of model simplification. If I do that, how off would I be from reality?

EDIT2: I've been doing further research and found something called a "worm like chain" model. I'm pretty fuzzy on what it means for the fiber, and I'm having trouble grasping how the fiber would move from an intuitive perspective; the visual for how I'd need to treat it isn't working, and the visual would help immensely with the actual modeling for the rest of the equations I need.

  • $\begingroup$ Can you give us more information about the specifics of the model you want to develop? Is this a computational model (like a simulation), or a mathematical model, for example? The persistence length is a statistically defined property, and so it is usually only important and meaningful in statistical models. When the chain length is much longer or shorter than the persistence length, the effect is usually that one or the other terms drop out of the equation. If they are about equal, then neither term would drop out. $\endgroup$
    – thomij
    Jun 5 '15 at 17:56
  • $\begingroup$ The model is statistical, and the issue I have is that I know the length of my polymer is about equal to its persistence length. I'm trying to get a more thorough understanding of what that means so that I can derive a more detailed model of the polymer fragmentation. $\endgroup$ Jun 5 '15 at 19:35
  • $\begingroup$ You should read en.wikipedia.org/wiki/Persistence_length - spaghetti is nowhere near its persistence length. $\endgroup$
    – Mithoron
    Jun 5 '15 at 20:04
  • $\begingroup$ Wikipedia refers to uncooked spaghetti. $\endgroup$ Jun 5 '15 at 20:59
  • $\begingroup$ If my answer doesn't give you enough information, you might also try Polymer Physics by Rubinstein and Colby for an introductory look at polymer statistics, or The Theory of Polymer Dynamics by Doi and Edwards for a more detailed and complete overview. $\endgroup$
    – thomij
    Jun 6 '15 at 4:28

The persistence length is a statistically defined property - in other words, it is defined by looking at many lengths of many polymer molecules over a long period of time. It is the length (L) at which a vector drawn tangent to the polymer is no longer correlated in space with another tangent vector at distance (L). In other words, it tells you how well the position of one section of a polymer is correlated to another section. Beyond the persistence length, you would not expect to find any correlation. Below it, you would expect correlations.

In this case "correlations" are measured by the cosine of the angle between the two vectors. When the angle is 90 degrees on average, the cosine is zero, and when it is zero degrees, the cosine is 1. So, for lengths beyond the persistence length, the amount of correlation is zero, and for lengths much less than the persistence length the amount of correlation is one. For lengths close to the persistence length, the correlation is between 0 and 1.

To give you an idea of how this works, take a look at the following illustrations:

Tangent vectors on a polymer chain.

The red lines represent tangent vectors along the backbone of a polymer chain, spaced at regular intervals. We can translate all of these vectors and put their origin at the same point, and then find the angle between them:

Tangent vectors translated to share an origin.

If we find another polymer chain, or watch this one for a little while and take another "snapshot" and do the same thing, we might end up with something that looks like this instead:

Another set of tangent vectors, translated to share an origin.

We can even take the same very long chain and find all of the vectors that are a distance 1 apart, 2 apart, and so on. If we do this for all possible distances over hundreds of polymer molecules, and average over many snapshots for a very long time (nano- to micro- seconds for a melt or something in solution), what we will eventually find is something like this:

Ensemble average of tangent vectors, translated to share an origin.

The correlations drop off quickly with length, then more slowly, until finally they are completely uncorrelated.

It is important to realize here that at any given time, for any given polymer, the tangent vectors could be almost anything. You wouldn't expect two adjacent bonds to have a very small angle between them, and in general you would expect more flexibility over longer lengths of chain compared to shorter lengths, but you would not be able to predict the shape or angles with any degree of certainty, other than to say that on average, beyond the persistence length the tangent vectors would not be correlated, and that the amount of correlation would decay exponentially with length.

The examples you gave of a stiff rod vs. cooked spaghetti are meant to demonstrate this intuitively, but your question is about what happens when the persistence length and the polymer itself are the same length. The answer is that the polymer will be somewhere between a very flexible piece of cooked spaghetti and a stiff rod. Imagine taking a piece of spaghetti and bending it into a loop, but not enough that it breaks. The circumference of 1/4 of the loop would be something like the persistence length, and the response of the spaghetti in the loop would be similar to a polymer that is about the same length as its persistence length. However, keep in mind that polymers are constantly in motion, and so "flexible rods" and "cooked spaghetti" are not great analogies in the first place. It's more realistic to think of them as constantly wriggling worms, or something like that. Statistical descriptions of correlation lengths and times are the best we can get, unfortunately.

Another concept that might help you visualize this is that of the Kuhn length - this is twice the persistence length, and is the length at which sections of polymer are uncorrelated with each other in space. Practically, it means that the motion of the polymer within the Kuhn length is irrelevant to the motion of the overall chain. On a large scale, the motion of the Kuhn "blobs" is what matters. In your case, the polymer is about as long as the persistence length, which means it is one-half of the Kuhn length. This means motions along the polymer are correlated, and so you can't ignore them (if they are important to your model.)

Practically, what this all means is that your model is probably going to be fairly complicated, because you can't average out or ignore terms that can be ignored in either of the extreme cases.

If you post some more details about your model, I could probably give you more specific information.

  • $\begingroup$ The main specific thing I'm trying to find is a way to determine how the polymer will arc: in detail, whether parts of it will do so semi independently of other parts, and statistically, the distribution how many degrees each arc will be. $\endgroup$ Jun 6 '15 at 17:57
  • $\begingroup$ The persistence length isn't really related to an arc angle in the way I think you mean. I am not sure that there is a well defined concept of arc angle for polymers at all - polymer dynamics are very random and the conformation of a single polymer at any given instant could be anywhere from linear to tightly coiled. How do you want to define the arc angle? $\endgroup$
    – thomij
    Jun 6 '15 at 23:34
  • $\begingroup$ The arc angle would be the central angle of the theoretical circle the arc would be a part of: eg a quarter circle would be a 90* arc. $\endgroup$ Jun 8 '15 at 1:30
  • $\begingroup$ How would you map the polymer backbone to a circle? What I mean is, over what length scale do you expect to find two dimensional arcs? I would be very surprised if you could define an arc in two dimensions over more than three backbone bonds. In that case, the distribution of arc angles would be the same as the distribution of the dihedral angle. It might help if you could explain what physics your model is intending to capture - I have a feeling that you have a conceptual picture of polymer dynamics that is much more orderly than how they behave in reality. $\endgroup$
    – thomij
    Jun 8 '15 at 3:08
  • $\begingroup$ I'm hoping I can at least simplify the model a little by saying that all the bends in the polymer are a linear combination of circular arcs. What I'm trying to find is basically a rough distribution of the angles of these arc elements. If there's a distribution (not just a mean, but an actual distribution) for these angles, then that would be absolutely perfect for what I am looking for. $\endgroup$ Jun 8 '15 at 14:56

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