Relationship between Radius of Gyration, Molecular Weight & Intrinsic Viscosity

That makes sense about using the equations instead of the raw data for the very high correlation.

Also, I just realized too that 1,4-polyisoprene in both dioxane and cyclohexane as well as 1,4-polyisobutadiene in benzene (a bad solvent for it) are the ones that have the greatest deviation from experimental and calculated Rg values with greater than 2 slope (after using corrected Flory Constant), and the Fox-Flory Equation is ideal for linear flexible chains, and these polymers are much more rigid due to their alternating double bonds along the polymer backbone, so everything makes sense now.

I learned so much today!

Hey Thanks a lot MJC,

Changing the Flory Viscosity Constant from 3*10^24/mol to what you used 2.5*10^23/mol made a huge difference and gives me so much more confidence in this analysis!! I can live with those results.

I tried to attach the data in an Excel spreadsheet with all polymers in their respective solvents and then calculated for various Molecular Weights using the K & a values from the article so you could look through the formulas but it would not let me so I instead had to attach it as a PDF. So if you have any questions about the equations, please ask (I basically just multiply the theoretical MW in left column by the KM^a from article in top header.

The top data set is my original calculations using the 3*10^24/mol Flory Viscosity Constant, while the bottom set is using the 2.5*10^23/mol you used. There is still some discrepancy with the Rg calculation for the 1-4 polyisoprene in cyclohexane being 2.91 times higher than th Rg from the article, but many of the others are all around a slope of 1. Even for a theoretical 500,000,000 Da molecule the R2 is still always very close to 1.0000. Does that make sense to you?

cheers mate! You don't realize how much this helps me....

My apologies MJC -- I've been busy for last few months and not on the forum as much.

Yes, I did get some useful results with my light scattering and viscosity detectors for the polymers. Very interesting indeed. Yes, hydrocolloids in a good solvent with very extended structures (eg. Mark-Houwink-Sakurada equations [n]=KM^a with the a exponent greater than 0.9 such as xanthan, carrageenan and sodium alginate) did give varying Flory-Fox coefficients not only for different polymer-solvent (aqueous) systems, but also for same polymer-solvent systems but at different MWs. The F-F coefficient was most constant when close to theta conditions (a=0.5) for different polymers and for different MWs, based on increasing the NaCl concentration in water to suppress ion effect of the polyelectrolytes.

I'm hesitant to guess about comparing results from different techniques. As I understand it, so far we've been using the same experimental data, and comparing the results of calculating R_g using different equations. Under these conditions it appears that in good solvents FF does predict a slightly lower a exponent, so at high MW the R_g value will be lower. For the case of PS in THF, see the comments on p626 of the paper. I must admit I haven't come across "kinetic energy corrections" before, and I don't know what they are, but it's clear that THF is recognised as a special case. Finally, you refer to "hydrocolloid polymers". Will the theory apply to these? Are they "flexible linear chain polymers"? Are they something other than simply hydrophilic polymers in aqueous solution? "Colloid" suggests particles in suspension, and I don't think FF would work for that.

Thank you for catching those errors, MJC. Before you corrected the Flory Viscosity equation, all the slopes were off so I did not consider looking for individual typos, but should have when these three polymers were not following along with the others.

OK, and I understand what you are saying about the polyisoprene behaving like a flexible linear chain still but having longer statistical segment lengths since it cannot rotate at the double bond.

Again, I wish to express my gratitude for your help with this and spending time to go through the data since I knew there was a chance my question would not have been answered as when I did a topic search on this I did not find anything very similar. I will not be using SEC-RALS-Viscometry for hydrophobic polymers but studied this article as a way to learn the principles and ensure the calculations work. Instead I will be running hydrocolloid polymers in aqueous and salt solutions so I hope that everything I've learned from this will translate over into the polar phase!

Ah, I see, you were calculating [η] and R_g from the equations in the paper. I was using the raw data from the tables. That could explain the good correlation. Perhaps it's a matter of Φ varying in certain circumstances. Or possibly even a misprint of the K value in one of the equations, especially if it's just one solution that's out. Thanks for the data, I'll look over it at leisure and come back with any comments. Glad I could be of some help.

First, I think there's a mistake in your value of Flory's constant; it should be an order of magnitude smaller. This ref
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-50532005000300017
quotes values of Φ from different theories with a range of 1.81-2.87 x 10²³ /mol (misprinted as 10^-23). I haven't been through all the data, but for polybutadiene and polyisoprene I get very good agreement (in those cases where both [η] and R_g values are given), using a value of 2.5 x 10²³ /mol, in the theta solvent dioxane. The agreement is less good (particularly for PIP) in the good solvent cyclohexane, and gets worse with increasing molecular weight. (This would not give R² = 1. Can you share your data?)
I don't know the derivation of your equation - whether, for example, it only strictly applies to theta solutions, in which [η] and R_g both vary as M^1/2 - but looking at the fearsome equations in the brazilian ref it seems that significant modifications are needed for good solutions - which may be interpreted as a varying "effective Φ". Your point about second virial coefficient then seems on the right lines; the theta condition [polymer, solvent, temperature] is that at which A₂ = 0, the strong interactions in good solvents complicate things.

Just curious - did you try it with your hydrocolloid polymers and did the F-F equation work?

You have R = 1/sqrt(6){[η]M/Φ}^1/3
If [η] = KM^a then R = const.{KM^a+1}^1/3 = K'M^(a+1)/3
In theta solvents, both a's are theoretically 0.5, and practically pretty close to this. In good solvents typically a([η]) is 0.7-0.75, implying a(R_g) just under 0.6, but in these examples a(R_g) is usually about 0.6 or just over.

I understand -- Yes, for the polymers in good solvents that show a higher slope (~1.3) when correlating the multi-angle light scattering results for Rg from article with Rg calculations based on the F-F equation also have slightly lower linear R^2 values which are corrected when changing linear to power regression.

So from this, would estimates of Rg from SEC-RALS-Viscometry using the F-F equation be lower than that obtained from MALS with Zimm Plot when the polymers are a) much higher MW and b) in a good solvent? Or any other scenarios where I might suspect deviations? And what would you say is the case of Polystyrene in THF since the F-F equation gave higher Rg results as seen by the slope lower than 1 yet THF is supposed to be a good solvent for polystyrene?  Thanks.

I'm afraid it's simpler than that. In these 3 cases you have incorrectly copied the K value for [η] from the paper. The values should be 0.111, 0.0197 and 0.107 respectively. Using these values gives slopes much closer to 1. (Should be the first thing you check if your calculations give odd-looking results.) So all that speculation about varying Φ, second virial coefficients etc....

You will find that generally the slopes are close to 1 for theta solvents, not so close for good solvents. This is because the F-F equation implies a(R_g) = (a([η])+1)/3, which is pretty closely followed in theta solvents but only approximately in good solvents. The result is that your plot for good solvents (e.g. PBD/CH) is noticeably curved, despite the high correlation coefficient.

And the double bonds don't make polyisoprene and polybutadiene rigid - they just make the statistical segment length longer. For high MW, they will behave as flexible linear chains. But polyisobutylene (there is no such thing as polyisobutadiene) has no double bonds - it is [CH₂CMe₂]_n.

MJC, can I ask you to give some more details on how the F-F equation implies a(Rg) = (a([η])+1)/3 since I want to understand this part of what you said better? Thank you again.