Comment pubs.acs.org/Macromolecules
Reply to “Comment on ‘Chain Entanglements in Polyethylene Melts. Why Is It Studied Again?’” V. M. Litvinov,*,† M. E. Ries,*,‡ T. W. Baughman,§ A. Henke,† and P. P. Matloka§ †
DSM Resolve, P.O. Box 18, 6160 MD Geleen, The Netherlands School of Physics & Astronomy, University of Leeds, LS2 9JT Leeds, England § DSM Ahead Materials Sciences R&D, P.O. Box 18, 6160 MD Geleen, The Netherlands ‡
Macromolecules 2013, 46 (2), 541−547. DOI: 10.1021/ma302394j Macromolecules 2013, DOI: 10.1021/ma400682z
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similarly be extended to include the previous data, which for completeness we now show here in Figure 1.
ecently, we published an article that, among other things, quantitatively determined the entanglement molecular weight, Me, for linear polyethylene (PE) chains in the melt by analyzing the Hahn echo sampled transverse NMR relaxation.1 In response to our publication, Saalwächter has written a Comment,2 in which several of our assumptions and conclusions have been challenged. In this Reply we will address each issue raised by the Comment. Saalwächter argues2 that our “central claim” is of an “increased Me”. We reported an NMR determined value for Me of 1760 g/mol and stated, “This value is in the range of previously reported Me for PE.” We therefore do not consider it our central claim that the entanglement molecular weight for polyethylene needs to be increased in light of our NMR results. Our main purpose was instead to examine new high molecular weight, low polydispersity PE melts using an exact NMR theoretical result3 combined with computer simulations4 and a rescaling approach5 to calculate a quantitative value for the entanglement molecular weight. We did say in our article that our NMR determined value was higher than the commonly accepted value for the rheological entanglement molecular weight, 1250 g/mol, and tried to suggest reasons for this difference. The list of possible explanations given in our publication was not meant to be exhaustive, as it presupposes our NMR work to be correct. Different techniques measure different properties and make different assumptions and approximations in their interpretations. Therefore, if we gave the impression that the central result from our work was that the entanglement molecular weight needs to be increased because of our NMR analysis, then the Comment from Saalwächter gives us the opportunity to rectify this. In the Comment Saalwächter calculated the Me as determined by neutron spin-echo spectroscopy to be 1670 g/mol and states, “This is in reasonable agreement with the rheological value...”, and we likewise consider our NMR determined value of 1760 g/mol to also be in reasonable agreement. We did compare in our article our measured Me with a previous result,4 where the NMR measured Me was found to be somewhat lower at only 1230 g/mol. This could have contributed to the idea that our central claim was of an increased entanglement molecular weight. In reality, though, our data were entirely consistent with that previous work, which was the intention for showing Figure 7b in our article,1 in which we plotted the results from both studies1,4 on the same graph. Figure 6 in our article1 can © XXXX American Chemical Society
Figure 1. NMR determined entanglement molecular weight as a function of the inverse of the polyethylene chain number average molecular weight for samples in the present study1 (closed circles) and from previously studied PE samples4 (open circles) of lower molecular weight. Solid line represents the result of a linear regression analysis: intercept = 0.9 ± 0.2 kg/mol; slope = 386 ± 8; the correlation coefficient = 0.99; the standard deviation = 0.64 kg/mol.
It must be noted that the apparent molecular weight dependence of Me shown in Figure 1 does not indicate that the molecular weight between entanglements is in reality a function of chain molecular weight. This is an artifact of our analysis, which was discussed1 in our publication. Therefore, the previous work is in reality consistent with our more recent work. The main reason for the lower value of Me reported earlier probably stems from the smaller range and much lower values of molecular weights examined. This made the extrapolation involved in that analysis, needed to calculate the entanglement molecular weight, less reliable. Our new samples with their far higher molecular weights gave us the opportunity to improve on that previous work. Continuing this theme of comparing the results from our recent article1 with that of the earlier work,4 Saalwächter criticizes the earlier use of the Carr−Purcell−Meiboom−Gill6 (CPMG) technique and argues further2 that the “previous fits cannot be considered reliable”. Here we stress that the results Received: May 24, 2013
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eq 1 of our article1 and how this evolves with time, which is what determines the result given by eq 3, are however being correctly calculated by our approach. But, it should not be concluded that we disagree with Saalwächter and his work on the orientation autocorrelation function; instead, our approach is effectively evaluating the integral of P2(cos α) over the time scale of our experiments, where angle α is the time-dependent angle that an intraproton pair vector makes with respect to the external magnetic field and P2(...) is the second Legendre polynomial. In the Comment it is stated that “It is an unfortunate weakness of this paper that the given, well-characterized samples were not investigated at different temperatures...”. We carried out the NMR measurements at 150 °C to be consistent with the previous4 NMR work. But most importantly in this respect is that the polyethylene samples have an end of melting at ∼135−145 °C depending on sample. Although the highest temperature that we can access with our benchtop spectrometer is 200 °C, the experiment at higher temperatures are not desired because of oxidations and chain scissions reducing molecular weight. Therefore, we have little room for maneuver when it comes to a useful and accessible temperature range. To quantify the entanglement molecular weight from our NMR data, we need to make use of a rescaling constant κ.5 This was calculated for polyethylene from a computer simulation4 and is used in eq 3 of our article. Saalwächter is correct to point out in the Comment that if this numerical value is incorrect, then the value for the entanglement molecular weight we calculate is similarly incorrect. In the computer simulation article only the dipole coupling between two protons in a CH2 group was considered, and so Saalwächter is again correct to point out that there are many more spins actually involved, which will affect the numerical value for κ. The interesting point to note though is that these extra interactions will increase the size of the interactions and therefore increase the magnitude of κ. In our analysis this would have the effect of directly increasing the magnitude of the parameter Me. This arises because the fitting procedure determines the value of the ratio κ/Me, from which we then calculate Me; recall eq 3 in our article.1 Therefore, this, though a correct point to highlight, will not have the desired effect of reducing the value for Me that we reported. It should also be noted that the contribution of the extra interactions is rather small (ca. 10−15%) as was previously shown by NMR relaxometry of mixtures of protonated and deuterated polymers.12−14 Nevertheless, we recognize this as an important issue to have been raised. Finally, we come to an extremely interesting objection from Saalwächter, who has demonstrated15,16 using MQ NMR that the distribution of residual dipole−dipole coupling constants in polymer networks is not consistent with that expected from a Gaussian distribution of chain end-to-end distances. Indeed, in theoretical work17 by Brereton and Ries it was also found that a Gaussian distribution of network vectors was not able to satisfactorily explain the double quantum buildup and decay signal from a polymer network. However, in this work on entangled linear chains1 we have indeed assumed that the chains on the entanglement length scale are described by a Gaussian distribution. The assumption of Gaussian statistics in polymer melts is widely used in the polymer community and is not peculiar to our publication. Furthermore, in previous work the NMR responses from various polymer systems18−20 have been successfully analyzed using this assumption. It is not clear
of our recent article in no way depend on those previous results. Nevertheless, as shown above they are consistent with each other. Therefore, we do not agree that the previous work is unreliable. It must also be stated that in section 2.4 of our article1 we did actually highlight the problems with the CPMG sequence for these samples. In the Comment, Saalwächter addresses our assumption of fixed entanglement points, which we make use of to analyze the transverse NMR relaxation signals. The idea is that for the polyethylene chains in this study the molecular weights are much larger than the entanglement molecular weight (Mn ≫ 100Me), and therefore the reptation time is far greater than the NMR experimental time (∼10 ms). So on the time scale of the NMR data, we considered the polymer melt to be a network formed by the entanglement points. Here Saalwächter is correct in that we should have discussed this description in more depth, explaining why we took that approach and its potential7−9 criticisms. Fixed entanglements was a simplifying approximation, which then allowed us to write down an exact analytical result for the transverse relaxation decay; see eq 3 in our article.1 This theoretical result gives a signal which has a very distinct time dependence that the data exactly matches; recall Figure 5 in our article.1 The theory only has one fitting parameter, namely the entanglement molecular weight, and as argued above, the value obtained is in reasonable agreement with that found from other techniques. In our article, when considering the apparent molecular weight dependence of the entanglement molecular weight, we did however state, “This is an interesting result which shows that even for these large polyethylene chains with Mn > 100Me our assumption of fixed entanglement points is not quite correct. The chains have more freedom to reorient on the time scale of the NMR experiment than this description gives.” Mathematically, it is possible to give the network chains used in our analysis a relaxation time.3 This would allow the entanglement points to be not fixed in space. The result is that this alters the time dependence of the NMR relaxation signal; see Figure 2 in the work3 by Brereton. This time dependence was not seen in our data, and we can therefore deduce a lower limit on the network vector relaxation time of 50 ms that would have been detectable from our data. An alternative way to model the entanglement points would be to allow them to fluctuate about a fixed point. This was considered in an earlier work by Brereton,10 where it was found that for isotropic fluctuations the analytic form of the relaxation signal was not changed by the inclusion of these fluctuations. It is really important to highlight that our fits to the data, as shown in Figure 5 of our article,1 are the most rigorous test of eq 3 ever to be presented. These high molecular weight chains with their correspondingly low density of free ends means that over 99% of the signal decay was determined solely by eq 3, whereas in previous work the free ends have always hidden11 the full distinct nature of the time dependence of that result. Equally, it must be stressed that Saalwäc hter has pioneered the determination of segmental dynamics by 1H multiple-quantum (MQ) NMR in entangled polymers, 7−9 in which the autocorrelation function C(t) determined by his work, shown in Figure 1 of the Comment, does directly challenge our simplifying idea of fixed entanglements. We are presently unable to calculate, by an exact method, the transverse NMR relaxation function for the C(t) presented in the Comment. Our work indicates that the numerical value for the integral in B
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how to reconcile this with the work by Saalwächter, and this does indeed remain2 a “still open problem”. We consider that our publication1 adds to this discussion, showing that Gaussian statistics are consistent with our NMR data from entangled polyethylene melts. As mentioned above, the analytic transverse relaxation function derived from the assumption of Gaussian statistics has a distinct form that the data clearly matches, resulting in a single parameter fit, the entanglement molecular weight, giving a value that is in the range determined by other techniques. This comparison of theory with data and the agreement with other methodologies means that at worst our work could of course still be incorrect, but our conclusions are most certainly not as stated by Saalwächter2 “a circular statement without factual basis”.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (V.M.L.); M.E.Ries@leeds. ac.uk (M.E.R.). Notes
The authors declare no competing financial interest.
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REFERENCES
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