Talk:Rubber elasticity

This comment pertains to an early version of the article. The text that it refers to is now in section 'Historical approaches to elasticity theory'. — Preceding unsigned comment added by Davidhanson471 (talk • contribs) 03:53, 3 December 2019 (UTC)

Arithmetic Problems

   This treatment suffers the difficulty of disagreeing with observation, namely that force (at constant temperature) =(Const)x r cannot change sign, as r, the end-to-end distance cannot be <0, while in practice rubbers are often strained in (linear) compression as well as in extension--e.g., tire treads, shoe soles, shock mounts.  The difficulty is not in the model but in the arithmetic.
   
  The treatment is based on an expression for the probability that the chain ends are r units apart as a function of r.  Apart from notation this expression is the same as Eq. 3.9 in Ref. 4 (Treloar, "The Physics of Rubber Elasticity")(where it I given as P(r), not P(r,n), as n is a constant).  Differentiation shows the expression to have a maximum (most probable value) at r = √(2nb2/3)--see Ref 4, Eq. 3.10 and Fig. 3.5(b).   As entropy S≃lnP, S likewise would have a maximum at the same value of r; and as force F≃-dS/dr, F will accordingly be positive or negative as r is greater than (extension) or less than (compression) its most probable value, as is observed in practice (Ref. 4, Fig. 5.6).

  In the wiki article, when ln(P(r,n)dr) is substituted for S, only the exponential factor is carried over; the quadratic factor, which would constitute a 2ln(r) term is dropped.  Inclusion of the entire expression would lead to the appropriate maximum in S and a F(r) function similar to Ref. 4, Fig. 5.6, in agreement with experiment.

  Although ref. 4, Fig. 3.5(b) shows the maximum of P, and hence S, and hence +&- F as in Fig. 5.6,  it comes up with (Eq. 3.22) essentially the same force expression as the wiki article and which likewise cannot change sign.  This seems to result from substituting an expression for the rectangular-coordinate volume element, dτ(=dxdydz) for the spherical-coordinate volume element (cf. Fig. 3.6) and regarding it as a constant; this drops the 2ln(r) term from the entropy expression (Eqs. 3.18 and 3.19), just as the wiki article did.
  Ref. 4 observe that its "result (3.19) shows the entropy to have its maximum value when the two ends of the chain are coincident (r=0)...".  Eq. 3.9 shows P(0) = 0, which would have the probability of the maximum entropy conformation to be zero.  This unlikely event, as well as the disagreement with observation, can be avoided by keeping track of the terms.

Debosley (talk) 21:10, 29 May 2014 (UTC)

Serious problems with this page:

I applaud the effort of D Hanson in updating this page with modern modeling. The problems I see are

1) lack of balance ... apparently Hanson's work is the only work that is not merely historical. References are to him, or before 1950. Really? Nobody else at all?

2) No distinction between general truths of rubber elasticity and the details found by Hanson for polyisoprene; e.g. no discussion of the ideal rubber

3) A serious error in that dealing with polyisoprene (?cis 1-4 ? it is not clear) the word 'crystal' never appears. It is generally agreed that the turn upwards in the stress-strain curve for this material at high strain and room temperature is caused by strain induced crystallization (SIC). In this article (and Hanson's papers) it is said to be caused by elastic deformation of the chain. This would happen in materials that do not crystallize but they generally fail before getting into this regime. It is commonplace that molecular modeling does not capture crystallization in temperature regimes where it is experimentally seen; the space and time limitations of even modern computers do not include the nucleation events. But this does not mean it does not happen. I do not understand why the author does not restrict his model to the regions of strain or temperature where crystals do not form.

4) The falling slope of the stress-strain curve at lower strains is associated with details of molecular motion. But at least some of it is simply due to using engineering stress when the sample is reducing in cross-section. Identifying deviations from neo-Hookean would be more relevant.

5) A special region 1a is described in molecular terms and vaguely indicated in fig 1 as below 100% strain. The references show it to be a specially stiff region at up to 5% strain. Is there any experimental evidence for this to exist? It seems a simple experiment to do.

I do not want to get into an editing war here, but I feel the current version does not serve our readers well. Clavipes (talk) 03:49, 21 September 2019 (UTC)

======================= RESPONSE ======================Davidhanson471 (talk) 16:05, 4 December 2019 (UTC) Thank you for taking the time to critique the article. With regard to problem 1 in your list that the article exhibits a ‘lack of balance…Nobody else at all?’, it depends on what you mean by ‘balance’. The original Rubber Elasticity page (before I edited it) appears unchanged in the section entitled Historical approaches…, at the end. In my opinion, this description provided the reader with a confusing and incomplete picture of elasticity and was of no use to either the scientist/ engineer or the casual reader. There were of course a number of previous polymer physics theories such as the so-called 3-chain model. These theories all had serious deficiencies. In the case of the 3-chain model and its variants, all of the network chains were assumed to be of the same length and oriented symmetrically with respect to the strain axis. These and other objections to ideal chain theories are discussed in a review article, cited as ref. 4. Interestingly, none of these elasticity models were mentioned in the original version of the Rubber Elasticity page. In my judgment, it didn’t seem appropriate to ask the reader to slog through an extensive critique of the previous unsuccessful theories when a modern successful theory was available- a theory benchmarked against experiments.

With regard to the second problem in your list having to do with ‘general truths’ and ‘ideal rubber’, I am not sure how to respond since I don’t know what is meant by these terms. The term ‘ideal’ is commonly used in reference to free polymer chains that obey Markov statistics. Chains that make up a rubber network are of course not free.

Problem 3 of your list is concerned with the omission of a discussion of strain-induced-crystallization (SIC) processes in strained rubber samples. The recent experimental papers[Toki (2002) and Tosaka (2007)] that I have read suggest that the source of the x-ray scattering remains controversial. The crystal structure is variously assigned as monoclinic, lamellar or ‘shiskabob’. Since the intensity of the x-ray scattering peaks do not correlate well with either the strain or the observed stress, I am perplexed as to why SIC continues to enjoy such widespread acceptance as the source of the elastic force. The earliest reference to SIC that I have come across is a 1942 paper by Dart in which it appears as the explanation for the upturn of the stress at high strains without any discussion. I have not run across the origin of this claim. My guess is that the x-ray scattering experiments provided a convenient escape from the failure of the 3-chain rubber model at high strain, i.e. it did not predict an upturn to the stress. However, since SIC is so well known, I would have no objection to including a discussion of it under the Experiments section of the article. At present, I just don’t know how to do that without getting into a lengthy adjudication of all the experiments. The argument supporting the assignment of the stress upturn to non-entropic bond stretching and deformation is as follows: (1) We know that a rubber sample breaks if the strain is high enough. (2) The reason it breaks is because bonds on the chain backbone or crosslink rupture. (3) State-of-the-art Quantum Chemistry simulations (Density Functional Theory with non-restricted basis sets) provide a good estimate of the bond rupture process; the chain rupture force is about 7 nN for C-C bonds and 1.5 nN for S-S bonds in sulfur crosslinks. The simulations also provide the force-extension model before bond rupture occurs. (4) When this force model is used in the network simulation code, quantitative agreement with experiment is obtained (Fig. 1) in the high strain region (including failure). There are NO FREE PARAMETERS in code for this strain region. (5) Since the regime II force model correctly predicts not only the stress upturn but also the failure point, it suggests that additional elastic forces such as SIC are unnecessary.

Problem 4 in your list claims that the simulation code is in error at lower strains because the elastic behavior is plotted as the engineering stress. Engineering stress was used simply because it is the most common convention in the literature and, presumably, the most familiar to the reader. The simulation cell is deformed by an affine coordinate transformation to impose a strain and the resulting reduction of the cross section is therefore included in the calculation of the stress.

Problem 5 is concerned with the ‘special region 1a’ depicted in Fig. 1. I’m not sure which reference your are referring to when you say that it is ‘specially stiff at up to 5% strain’. Ref. 4, Fig. 11 on page 330 shows that it extends to a strain of about 25%. This figure also shows experimental data and the theory and experiment are in very close agreement. It was difficult to show this in Fig. 1 of the article because the y axis had to be large enough to show the high failure stress.

I am certainly open to suggestions to make the article clearer. It was a struggle to try to make it useful to the engineer/ scientist and also be somewhat accessible to the casual reader. Davidhanson471 (talk) 22:38, 2 December 2019 (UTC)