Talk:Ideal chain

Ideal mix
If anyone has an example of a polymer whose monomers form an ideal mix with its solvant, or an experiment where a polymer behaves in some way or the other like an ideal chain, that'll be great.ThorinMuglindir 10:41, 29 October 2005 (UTC)

Hey ThorinMuglindir. A solvent in which a polymer displays ideal behavior is typically called a theta solvent. Generally if you have a solvent that is a fluid made up of the monomers of the polymer you get what is called screening of the excluded volume effects and the resulting behavior is ideal (ideal simply means we can ignore the excluded volume interactions and hence model the polymer as a random walk). In terms of the Helmholtz free energy the coefficient of the first term in the virial expansion is zero since tau=0 or T=theta. Practically there is no real ideal solvent since the temperature can never be exactly theta, but if T is close enough to theta the thermal blob size can be much greater than the typical size of the polymer and hence the behavior is ideal (all chains look ideal if they are below the thermal blob size). I think a typical example is polyethylene and polystyrene but don't quote me on that. I think a qualitative explanation of what makes a solvent an ideal solvent is probably more important since it would be hard to get data to add a graph in order to truly convince people that it is a reasonable model (although I guess a reference could be given to a text where it could be found). Here and in the polymer physics article somebody should put in the Helmholtz free energy for the chain...


 * many interesting stuff in your intervention, however I think there might be some confusion between excluded volume effects that happen between neighbouring polymers (thermal blob) and excluded volume effect between the very monomers of the single polymer, which the worm-like-chain model tries to account for. Nevertheless, I'd say that such a passage as "A solvent in which a polymer displays ideal behavior is typically called a theta solvent" has to be included in the article, which I'll do in a couple of days. maybe I can add more if I get to understand what you meant, after all it could be that it's me who's being confused. Stacy333 (talk) 14:50, 22 May 2008 (UTC)

Hi

There is an sever Error in the Artikel The End to End distance is Normal distributed, but the End-to-End Vektors are NOT! There you have to take into account, that the Ammound of vectors with a certain end to End distance R is Proportional to the Survace of an Sphere with the Radius R so the Korrekt Formular is

$$ P(R)=\left ( \frac{3}{2 \pi N l^2} \right )^{3/2}e^{-\frac{3 \vec R^2}{2 N l^2}}$$ or $$ P(\vec R)=\frac{\left ( \frac{3}{2 \pi N l^2} \right )^{3/2}}{4\pi R^2}e^{-\frac{3 \vec R^2}{2 N l^2}}$$ [[User:Patrickruediger, 10:04, 7.December 2006


 * Hi,I used to be Thorin Muglindir in the past, now I'm just little Stacy. Hmmm, if my former incarnation among you earthlings made such a mistake, it needs to be clarified. However, it seems to me that the probability associated to the end to end distance ought to have the dimensionality of the inverse of a length ($$m^{-1}$$), while the probability density associated to the end to end vector in 3d space ought to have the dimensionality of the inverse of a volume ($$m^{-3}$$). After correcting them, the formulas that Patrick comes up with appear to not have the correct dimensionality... I think it would be correct that the End-to-End Vektors is Normal distributed, but The End to End distance are NOT! but I'm afraid you somehow got the two mixed-up mein Herr, while the article is correct (at least its formula has the right dimensionality)... I'm not sure though, I will have to think about this although my physics degree is located a couple more years in the past than when I used to be Thorin Muglindir Stacy333 (talk) 14:50, 22 May 2008 (UTC)

Probability distribution
It may be worth noting that the probability distribution given in the article is only true for the end-to-end vector in the thermodynamic limit. The exact solution is dimension-dependent and can only be given as a series solution. See Yamakawa 'Modern Theory of Polymer Solutions' (1971, Harper and Row) for the complete derivation. ScottRShannon 02:06, 24 April 2006 (UTC)
 * The thermodynamic limit thing is mentioned in the article from my former incarnate Stacy333 (talk) 14:50, 22 May 2008 (UTC)

issue corrected
There was a small issue with the change in thermodynamical ensemble at the very end of the article. In the "fixed force" ensemble, it's a mean end-to-end vector, but it's not a mean force, the force is fixed by the operator as $$ \vec f_{op} $$. I had to come back with my previous account, since stacy can not (yet?) upload images. The image also had to be corrected in the same way. Sorry about that mistake from a few years ago. ThorinMuglindir (talk) 15:34, 22 May 2008 (UTC)


 * had to correct again because I had created a consistency issue, having defined $$ \vec f_{op} $$ as the opposite of $$ \vec f $$ in an earlier chapter. Everything should be consistent and correct now. The thing is, when you go for the length reservoir approach, you assume that force is constant and does not fluctuate.ThorinMuglindir (talk) 18:45, 22 May 2008 (UTC)

Constant |r| ensemble
The third ensemble is missing: See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1303618/pdf/3418.pdf, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.34.3486  — Preceding unsigned comment added by 129.69.120.91 (talk) 07:25, 10 December 2018 (UTC)

Correlations
"However, since the components of the vectors $\vec r_1,\ldots ,\vec r_N$ are uncorrelated for the random walk we are considering,"

This is completely wrong. The components are correlated, since |rᵢ| is fixed. The correlation disappears only in the limit of infinitely many links. --Ilya-zz (talk) 10:16, 17 August 2020 (UTC)