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A proton NMR investigation of crosslinks and entanglements in polyethylene networks and star polymers
Polymer Gels and Networks 5 (1997) 285-305 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 0966-7822/97/$174il PIlz SO!%-7822(97)00004-X
ELSEVIER
A proton NMR investigation entanglements in polyethylene polymers M. E. Ries,” M. G. Brereton;
of crosslinks and networks and star
P. G. Klein”* & P. Dounis,bt
“IRC in Polymer Science and Technology University of Leeds, Leeds LS2 9JT, UK bIRC in Polymer Science and Technology, University of Durham, Durham DHl 3LE, UK (Received
2 August
1996; revised
17 December
1996; accepted
18 December
1996)
ABSTRACT This paper investigates the ability of proton transverse NMR relaxation to study the structure and dynamics of polyethylene melts. Commercial linear samples of similar M,, and various M,,, have been examined, together with samples crosslinked by electron beam irradiation. Apart from the linear sample of lowest M,, all the free induction decays (FIDs) are very similar and this is attributed to the presence of free chain ends. The magnitude of this problem is illustrated by generating theoretical FIDs from networks of different crosslink density and reptating linear chains of different entanglement spacings, with and without free chain ends. A new result for the FID using the reptation spectrum is presented and used to fit data from the linear polyethylenes. This yields an “NMR average” molar mass, similar to the accepted value and provides information on a sample’s polydispersity. Three-arm polyethylene star polymers, of low polydispersity and various arm lengths, have also been studied. The large variation in the chain dynamics from the core to the free end of the star arm is not reflected in the FIDs, which can all be well-fitted by double exponentials. This highlights the importance of specific isotopic labelling in studying such systems. 0 1997 Elsevier Science Limited
1 INTRODUCTION Commercial polyethylene samples contain protons at all locations within the material. A transverse proton NMR measurement thus records the * Author
to whom correspondence should be addressed. t Present address: Exxon Chemicals Ltd, PO Box 1, Milton OX13 6BB, UK.
Hill, Abingdon,
Oxfordshire
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M. E. Ries et al.
cumulative signal from the different locations and presents an ensemble average. This complicates the interpretation and sets limits on what can be uncovered by proton NMR. In this work a range of commercial samples and specially synthesised low polydispersity star polymers are investigated to ascertain the extent to which proton NMR can reveal melt structure and dynamics without the use of specific labelling. The analysis of small linear chains undergoing fast dynamics is relatively simple, since the proton transverse free induction decay mainly displays a single exponential.’ However, many commercial polymeric systems can generate complex non-exponential decays.2 One of the main problems encountered in NMR analysis of polymer melts has been interpreting these signals in terms of structural and dynamical properties. Until recently these decays have been described by empirical formulae, with much effort being made to attribute the phenomenological terms to molecular correlation times, entanglement densities, crosslink densities and gel fractions. ‘-5 A framework to interpret the proton NMR transverse decay for a polymer chain in a variety of systems, ranging from short Rouse chains’ to been introduced into the entangled melts’ and networks,8 has recently literature. These techniques have already been employed to investigate monodisperse linear chains, including poly(ethylene)’ and poly(ethylene oxide).‘” In this work the investigation is extended from these relatively ideal systems to a range of complex structures; including highly polydisperse entangled linear melts, networks formed by electron irradiation and low polydispersity star polymers. The networks formed from the electron irradiation of poly(ethylene) melts illustrate the difficulty in distinguishing between crosslinks and entanglements by proton NMR. The abundance of free pendant chains and other mobile constituents formed by main chain scission mask the information in the transverse relaxation function concerning entanglement and crosslink density. The signal from a model theoretical network with and without free ends is then used to illustrate the potential of NMR to measure crosslink density and the problem introduced by “rogue” free ends. A new result for the free induction decay calculated from the reptation spectrum is presented and employed to interpret the linear entangled data. This gauges how sensitive proton NMR is to a sample’s polydispersity. A theoretical model is then used to demonstrate the problem induced by free ends. These peripheral segments of the reptating chain undergo fast reorientation and with this comes a gradual decay in the transverse relaxation. This again hides the detail concerning the more rapid NMR decay from the central reptating segments.
Proton NMR investigation of polyethylene melts
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Star polymers constructed at the University of Durham, which form the precursor to model networks, are also investigated. These offer a wide distribution of correlation times, from Rouse-like ends to highly constrained reorientation at the star’s core. These, thus, indicate the sensitivity of proton NMR to a specific spectrum of molecular relaxation times. This paper thus points out the strengths and weaknesses of proton NMR and sets out an analytic approach to interpreting the transverse relaxation from polymeric samples.
2 EXPERIMENTAL 2.1 NMR measurements All the NMR measurements were made on a Chemagnetics CMX-200 spectrometer, operating at 200 MHz for protons. The Levitt-Freeman” modification of the Carr-Purcell echo train was employed:
This self-compensating sequence ensures that every second echo is correct, providing the diffusional processes are not significant. Full quadrature phase cycling was employed, with the (7c/2) pulse lasting 2 ps and the dephasing time T typically l/2 ms. A total of n/2 = 200 points was acquired, at the peak of every second echo, with the resultant free induction decay being the average of 64 accumulations. 2.2 Sample details Three samples consisted of linear polydisperse chains with a similar number average molar mass, M, = 25 000 g/mol, but with different weight average molar mass M,,,. Two samples were spun poly(ethylene) monofilaments, which had been electron-irradiated at room temperature in a vacuum, one receiving a dose of 6.0Mrad and the other 3.5 Mrad. The details of sample preparation can be found in a previous paper.‘* The above information is summarised in Table 1. Three-armed polyethylene stars were produced by quenching living polypentenamer with 1,3,5benzenetrialdehyde and hydrogenating.13 The
288
M. E. Ries et al.
Linear Name
and electron
M, (before
AL7050 UN2912 HO20 P2 P5
branch lengths Table 2.
TABLE 1 beam crosslinked
treatment)
polyethylene
M, (before treatment)
glmol
glmol
24 24 26 28 28
63 000 221000 450 000 115000 115 000
000 000 000 000 000
were varied
and the three
samples
samples Treatment
none none none 3.5 Mrad in vat 6.0 Mrad in vat
investigated
are listed in
3 THEORY 3.1 Transverse
NMR relaxation
Dipolar interactions of the proton pairs, which are attached to the polymer backbone, cause the dephasing of the transverse nuclear magnetic components. Experimentally, the relaxation of the transverse magnetisation is recorded as the free induction decay G(t). This function is sensitive to both the correlation times of the various chain segments and the melt structure. A scale invariant model of the polymer chain for the NMR properties was first envisaged by Cohen-Addad14 and later developed by Brereton:” several monomers N, are connected together to form a statistical unit b with the dipolar NMR interaction averaged over this length scale. The
The
star
Name star 1 star 2 star 3
TABLE 2 polymer molar polydispersities
mass
M, (g/m4
MJM.,
49 800 62 800 95 300
1.42 1.54 1.54
and
Proton NMR investigation of polyethylene melts
289
Fig. 1. The hierarchical model for the NMR properties of a polymer chain governed by two distinct modes of relaxation. N, monomers are connected together to form one b unit, with a further Nh b segments forming a vector R.
transverse
relaxation function G(t) for the chain is then specified by15 G(t) = (co, [ +
I’ (3 co? &‘,- l)]) 0
where 0, is the angle between the statistical unit b containing the NMR probe and the magnetic field (see Fig. l), Ab is the resealed dipolar interaction constant corresponding to the statistical length b given by
with y the gyromagnetic ratio for a proton, d the proton-proton distance within the proton pair and p0 the permeability of a vacuum. The (. * -) in eqn (1) indicates an averaging of the NMR active statistical segment over all available conformations, in the time interval 0 to t, subject to any constraint that the environment imposes on the bond, e.g. crosslinks or entanglements. It is through this averaging that the NMR probe is sensitive to the polymer chain environment. The problem posed by the evaluation of eqn (1) has been treated in a series of papers.C8,‘5 For completeness the main results will be summarised in the appropriate sections.
M. E. Ries et al.
290
3.2 Scale invariance Entangled polymer chains in a melt form a hierarchical structure of dynamics, with Rouse dynamics occurring on a small scale and reptation on a larger one. We have shown that when the local dynamics are fast A,z, << 1, where z, is the correlation time of bond b, it is then possible to rescale further7 the NMR interaction onto the slower, larger scale motion of the chain. For example, consider the chain as a series of vectors R each formed by connecting Nb statistical units b together as illustrated in Fig. 1. In the regime of fast local level dynamics the transverse relaxation no longer depends on the rate of reorientation of these statistical units, but only on the motion of the larger scale segments R. This approximation is useful since it renders the precise form of the local dynamics unimportant, simplifying the NMR calculation and allowing a coarser grained view of the polymer molecule. The dipolar constant rescales further to AR, where7
The transverse
relaxation G(t) =
where
is then determined
(cos[F I’(3 cos2OR- l)]) 0
8, is the angle the vector
4 RESULTS 4.1 Poly(ethylene)
from
entangled
R makes
AND linear
with the magnetic
(4) field.
DISCUSSION
chains
and networks
Much research has gone into investigating the effects of electron beam properties of structure and mechanical irradiation on the This bombardment of high energy electrons results poly(ethylene).12.1”2’ in crosslinking and chain scission, creating a complex micro-structure. The final network configuration might contain pendant chains, trapped entanglements and closed loops. Ideally, one would like to know how all these influence the macroscopic properties of the sample. A first step towards this goal would be the determination of the melt micro-structure. Work2’,” carried out at the University of Leeds on poly(ethylene) networks using the Edwards-Vilgis2* “ Slip Link Model” to interpret force
Proton NMR investigation of polyethylene melts
291
extension data revealed that entanglements were quite loose and, therefore, distinct from permanent chemical crosslinks. The question is whether this difference would be revealed in an NMR experiment. A theoretical model will, thus, be employed to investigate the ability of NMR to distinguish the dephasing of protons held between entanglement points from those connected by crosslinks. The main objective of this work is to examine the use of proton NMR to analyse network structures. To this aim several linear poly(ethylene) samples of similar number average molar mass are investigated using transverse NMR. Two samples have been subjected to different amounts of electron irradiation with a further three being untreated, but having various polydispersities. This allows the direct comparison between networks with expected distinct crosslink densities and those with the linear entangled chains, to give an insight into whether NMR can distinguish between permanent and temporary crosslinks. Measurements of the transverse relaxation on the polydisperse untreated samples provide access to whether NMR can be employed to determine molecular weight (m.w.) distributions. It will be theoretically shown that the transverse relaxation function for a reptating chain is expected to display an approximately linear m.w. dependence as opposed to the weaker log(m.w.) found’,” for a Rouse chain. Whether this is sufficient to reveal the distribution of chain sizes in a poly(ethylene) melt will be indicated by the NMR measurements on the untreated samples. This work builds on a previous paper’ by Brereton et al., where well-characterised linear poly(ethylene) chains were investigated. From measurements of the transverse magnetisation and the employment of recently developed theories it was possible to observe both Rouse and reptation dynamics. This work also determined for poly(ethylene) the proton dipole coupling constant A6 = 5606s’, the fundamental Rouse relaxation time r, = 1.49 X lo-‘s (at 149°C) and the size of a Rouse unit 252 g/mol. These parameters found from the analysis of monodisperse linear chains are essential to the investigation of uncharacterised melts. The transverse decays from the poly(ethylene) samples, which show a range of polydispersity and network fractions, corresponding to Table 1 are displayed in Fig. 2. The first thing to notice is the similarity between the results. Apart from the AL7050 sample, which has the lowest polydispersity, the data lies on approximately the same curve. There are slight differences in the long time data. The transverse relaxations fan out, with the more rigid melts decaying slightly faster. For example material P5, which underwent the most electron irradiation and has a significant gel fraction ofl’ 0.55, is below the others in Fig. 2, with the linear melt with the greatest polydispersity being next.
M. E. Ries et al.
I
z
P5 AL7050
-3
-4
time/s
Fig. 2.
Transverse NMR relaxation of various poly(ethylene) lines through the data are fits to bi-exponential
samples, see Table functions.
1. The
Apart from this minor distinction an entangled melt of linear chains is essentially indistinguishable, as far as NMR is concerned, from a network. Indeed it is possible to fit the data quite adequately using a phenomenological decay Gbi-exp(t), see Fig. 2, Gbi_,,,(t) = a exp(-at)
+ (1 - a>exp(-@)
(5)
where a, cz and p are fitting constants. This reveals that the various forms of relaxation undergone by the different components within the melt become translated by the NMR experiment into a rigid and flexible component. Information about polydispersity, crosslink/entanglement density and chain scission has been lost in the resultant signal, as all the decays are similar. A major restriction on the ability of proton NMR to decipher melt configuration is the abundance of free ends in a melt. These dangling ends, or even small segments that have been cut from the main chain by electron bombardment, are highly mobile. Fast reorientation, as anticipated with these components, would generate a slowly decaying signal. Without labelling, i.e. the specific placing of NMR active nuclei at known locations, NMR measures the summated signal from all the protons in the
293
Proton NMR investigation of polyethylene melts
sample. This essentially averages the decay over the various motions undergone in the melt. The result is that any slow reorientating chains with correspondingly fast decaying signals are hidden by the more gradual decay of the mobile constituents. The information of interest, corresponding to reorientation restricted either by entanglements or chemical crosslinks, is lost in favour of these freer chains. The magnitude of this problem will be illustrated in the following two sections. 4.2 Theoretical
FIDs from crosslinked
and entanglement
networks
The more mobile free ends significantly reduce the potential of NMR to investigate uncharacterised melts. This can be highlighted by a theoretical example in which the signals from three model networks, of different crosslink densities, are contrasted with those from the same networks containing free ends. Each network is formed by crosslinking 10 segments of equal molar mass, M,, together as shown in Fig. 3(a). To disturb the model network chain, by the inclusion of free ends, the peripheral crosslinks are released as in Fig. 3(b). To generate the theoretical decay for the model networks it is assumed that the crosslink points are essentially fixed in space. This represents the effect a network would have on its constituent chains. A chain in the network is unable to diffuse far from its starting point without becoming hindered by the junction points. The chain is now pictured as a series of vectors R, recall Fig. 1, that link successive junction points together. The relaxation time for this vector is very large in comparison with the time scale of the NMR experiment. In this regime, known as “frozen” bond
Fig. 3.
Showing:
(a) network
chain
with no free ends; ends.
(b) network
chain with two free
294
M. E. Ries et al.
dynamics, the signal becomes insensitive to the time. The solution to eqn (4) in this regime isl’ Gmodel (t) = Re{(l - 2iA,t)-“2(1
vector
R's correlation
(6)
+ iA,t))l>
network
where AR is the resealed interaction strength crosslink density through (compare with eqn (3))
and iV, is the number
of Rouse
units between
corresponding
crosslink
points
to
the
given by
with mousebeing
the molar mass of a Rouse unit = 252 g/mol for poly(ethylene). Work by Kimmich et ~1.~ revealed that the outer entanglement lengths of a reptating chain behaved, as far as NMR is concerned, in a similar fashion to a Rouse chain. This was later confirmed by Ries et d.” in studies of poly(ethylene oxide) linear chains. Thus, for the case of the network with free ends the signal generated from these mobile segments can be approximated by that expected from a Rouse chain of length N,. The above signal becomes modified to G TT,2Lidr(t) = 2 Re((1 - ZA,t))“*(l
+ i&t>-‘1
+ 4 exp
-6A2,r,
ln(N,) t n
where r, is the fundamental Rouse relaxation time, which for polyethylene at 149°C is 1.49 X lo-’ s.~ The second term in the above equation corresponds to the expected signal from a Rouse chain as predicted and measured in work by Brereton and co-workers9~“’ Contrasting the signals generated by the above equations, Fig. 4, reveals how the information about M, is concealed within the chain end decay. The information corresponding to the crosslink density is found in the long time data. The inclusion of more mobile free ends, with their correspondingly slow decaying signals, covers these details. Information is still seen about the crosslink density in the signal from the “imperfect” network, but in comparison to the perfect network it is greatly reduced. To observe motion corresponding to that between crosslink points the chains need to be selectively labelled so as to remove the influence of the rogue free ends. We will now investigate the theoretical FID from a reptating chain in a
295
Proton NMR investigation of polyethylene melts time/s 0.02
Fig. 4.
Theoretical
transverse
0.03
0.04
0.05
relaxations from networks of different crosslink densities, with and without free ends.
melt. The transverse decay from a 25 OOOg/mol chain with labelled peripheral segments will be compared to one with unlabelled (i.e. contains no nuclei that contribute to the NMR decay curve) outermost segments, see Fig. 5(a, b). To achieve this a theoretical decay from a reptating melt is required, this is derived in the appendix. In the regime of fast large scale reorientation
(5
r, << t ) the relaxation is specified by
a)
Fig. 5.
Showing: (a) entangled
chain; (b) entangled
chain with unlabelled
ends.
M. E. Ries et al.
296
time/s 0
Fig. 6.
Theoretical
0.02
0.01
0.03
0.04
0.05
transverse relaxation functions for reptating 25 000 g/mol unlabelled ends for different entanglement densities.
The expression (10) generates the unlabelled ends. To include these modified as before, see eqn (9), to
decay for a reptating the above relaxation
G,,ptation (t) = M -M2M’exp[ -tF tfrre ends f-
chains,
with
chain with function is
Azr,gln(g)] C
P
2M, M
The signal generated by the entangled chain experiencing different entanglement densities Me (centred on the M, for poly(ethylene) of 2000 g/mol)‘” with unlabelled ends is shown in Fig. 6. The effect of free ends can then be compared in Fig. 7. The entanglement molecular weight dependence of the reptation spectrum survives the inclusion of Rouse-like free ends. In the unirradiated samples the entanglement densities are, of course, the same, the value being a characteristic of poly(ethylene), and this explains the similarities of the data, recall Fig. 2. 4.3 Crosslinks
vs entanglements
It is clear from the above two theoretical analyses that entanglements are ostensibly distinct from crosslink points in an NMR experiment, compare Figs 4 and 6. This will be true until the motion between the entanglement points becomes so sluggish that it appears frozen on the NMR time scale. Then the reptating signal will be identical to a network with a crosslink density equal to the entanglement density.
297
Proton NMR investigation of polyethylene melts time/s
-4 -5
i
-6 I
Fig. 7.
Theoretical transverse relaxation functions for reptating 25 000 g/mol different entanglement densities together with free ends.
chains
for
Inclusion of any free chains or imperfections would reduce the maximum molar mass of a reptating polymer that would be discernible from a crosslinked network. In Fig. 8 the signal from a 100000 g/mol reptating chain with unlabelled ends is compared with that from a model network. For a correct comparison the crosslink density has been set to the poly(ethylene) entanglement density of (=2000g/mol). The two transverse relaxations follow each other until they have decayed to about 5% (eP3) of their starting values. It is, thus, anticipated that these slight differences will not be discernible for a less model system.
time/s 0.02
Fig. 8. Theoretical transverse relaxation from 2OOOg/mol compared with a reptating chain 100 000 g/mol and entanglement
0.03
a model network of crosslink density with unlabelled ends of molar mass density 2000 g/mol.
M. E. Rim et al.
298
4.4 Reptation
fit to the linear un-irradiated
poly(ethylene)
samples
We now assess the predicted reptation relaxation on the untreated poly(ethylene) samples. The first experimental result to be investigated is the sample AL7050, which has the lowest polydispersity. The data will be fitted to an equation similar in form to (11) as
The value of Me is set at the entanglement density of poly(ethylene) =2000 g/mol. The free ends have been given a transverse relaxation for a Rouse chain of an entanglement length, as suggested by Kimmich et u!.,~ but modified by a factor cz. The ends are expected to reorientate slower than that of free chains, simply because they are tethered to a reptating polymer. The parameter (Yallows for this and gives an insight as to how distinct the free ends are to Rouse chains. It is thus anticipated that LY2 1. From the free fit, see Fig. 9: (Y= 2.89, which is of order unity as 1.0
0.8
0.6
0.2
0.1
Fig. 9.
Reptation
0.2
time/s fits to linear chain data.
0.3
Proton NMR investigation of polyethylene melts
299
expected and M = 19 400 g/mol with this being close to the number average molar mass of the sample 24000 g/mol. The closeness of the fit and the reasonable nature of the parameters supports the use of the reptation spectrum. It is interesting to ask what effective molar mass is required to fit the other linear samples. The difference between the AL7050 material and the others is essentially the molar mass distribution. The above values can thus be used, only leaving one free parameter M. The HO20 sample is fitted to the above expression (12) in Fig. 9, but with the previous parameters, other than M, being employed. This gives a value for M of 27 000 g/mol. Again the fit is reasonable and as expected M is slightly larger than the number average molar mass expressing the wider molecular weight distribution. 4.5 Polyethylene
stars
As part of the work being carried out in the IRC at the University of Leeds towards the ability of NMR to interpret network dynamics several poly(ethylene) stars have been made in the synthesis group at the University of Durham. The stars used in this analysis form the precursors to model networks. Stars exhibit markedly different reorientation to that of linear chains. They are unable to reptate due to the core of the star acting like an anchor in the melt of stars. A theory put forward by Ball and McLeish proposes that the star reorientates by “breathing modes”,24 the end of an arm has to retract along its length before protruding out into its surrounding matrix to adopt a new random conformation. For the stress on segments deep within the star (towards the core) to relax, it requires the complete contraction of the arm. This event is highly unlikely with the result being that the core is essentially “frozen”. The star polymers offer a unique distribution of correlation times, ranging from Rouse-like for the peripheral segments up to locked in orientations at the core. This presents an opportunity to see how capable NMR is in differentiating between various mechanisms of reorientation. These results can then be directly contrasted with the signal from linear chains of comparable molar mass to a star arm. This indicates the ability of proton NMR to distinguish different relaxation mechanisms and to analyse uncharacterised melts. The signal from the poly(ethylene) stars can be found in Fig. 10. Stars by nature of their junction point have fewer free ends per chain (arm). This coupled with the anticipated spectrum of relaxation generated by the tethering of the core might be expected to give rise to a characteristically different decay. However, this is not so. As with the poly(ethylene)
M. E. Ries et al.
300 0
-1
2 13 ,e
-2
-3
-4 U
time/s Fig. 10.
Transverse NMR relaxation of various poly(ethylene) stars and the linear HO20 sample. The lines through the data are fits to bi-exponential functions.
network results the data is reliably fitted by a bi-exponential function identical in form to eqn (5), see Fig. 10. Again the summation of the signal over all the proton pairs, constrained to mobile, results in a two component fit. Instead of being able to decipher the nature of the spectrum of relaxations undergone by the star, it is possible to visualise the star as equally rigid inner segments with a flexible free end. This loss of information is highlighted when star 1 is compared with a linear chain of similar number average molar mass to its arm in Fig. 10. This shows the lack of distinguishing features present in the signal for the star. If it is nearly impossible to differentiate between star polymers and linear chains using proton NMR then it cannot be expected to characterise unknown melt micro-structures. This again reminds us of the importance of labelling if information is to be learnt from transverse NMR measurements. 5 CONCLUSIONS In this paper the attention was focused interpret structure and dynamics from
on the ability of proton NMR to commercial polymer melts. This
Proton NMR investigation of polyethylene melts
301
possibility was investigated both theoretically and experimentally. Several various linear polyethylene samples, some of which had undergone intensities of electron beam irradiation, were observed using the transverse relaxation decays. This highlighted the problems encountered when examining unknown melt micro-structures. Previous force extension analysis performed on the same samples had revealed information about entanglement and crosslink densities and even the freedom of entanglements to slide along the chain.” However, NMR found it difficult to even distinguish between these samples. It was proposed that the different environments, reptating melt or networked chains, were concealed by the presence of dangling chain ends. A model was constructed to illustrate the problem imposed by Rouse-like free ends in the network. Two networks were compared, both of these had a well-defined molar mass between crosslinks, but one included dangling peripheral sections. At long times, Fig. 4, it was possible to distinguish crosslink densities for the perfect network. The inclusion of the freer more mobile end sections generated a slowly decaying signal that reduces the discernible information about crosslink density, recall Fig. 4. The untreated polyethylene samples had various polydispersities. The reptation time depends strongly on the molar mass of a chain. It was expected that these different distributions of chain length would be revealed through the decay. However, the transverse relaxations from these samples, apart from the sample of lowest M,, were very similar. Also, the signals could be well-fitted by a bi-exponential, indicating the lack of information revealed in the decay. The reptation spectrum was employed to calculate the expected signal from an entangled melt. This was then used to fit the signal from the linear chains. From this it was possible to measure an NMR average molar mass of the chain, which compared well to the accepted value. To weigh the ability of NMR to distinguish crosslinks from entanglements the theoretical signals from a network and a reptating melt were compared. The crosslink density was set to the entanglement density. This then allowed direct comparison between the influence of entanglements and crosslink points. It was clear that until the reptating chain became too large, lO’g/mol for poly(ethylene) at 14972, the motion of protons between entanglements is distinguishable from that between crosslinks by NMR. As part of preliminary work on model networks, a selection of polyethylene stars have been studied. These stars exhibit markedly different molecular relaxation to linear chains, stemming from the anchored core that disallows reptation. Rheological measurementsz5 have verified the theoretically predicted “breathing modes”, with this technique
M. E. Ries et al.
302
being readily able to distinguish the stars from linear chains. The NMR transverse decays for the polyethylene stars were bi-exponential in nature, recall Fig. 10. This makes it impossible to assign or confirm any form for the spectrum of relaxation modes. A linear chain with molar mass comparable to the arm of a star was experimentally contrasted with these results. Here it was found that the two signals were similar, highlighting the weakness of proton transverse NMR to investigate polymer melts without the use of specific labelling. We are currently pursuing this, studying selectively deuterated polybutadiene star polymers.
ACKNOWLEDGEMENT We wish to thank manuscript.
Professor
W. J. Feast
for his helpful
comments
on the
REFERENCES 1. Charlesby, A., Radiat. Phys. Chern., 39 (1992) 45. 2. Cohen-Addad, J. P. & Dupeyre, R., Polymer, 24 (1983) 400. 3. Bremner, T. & Rudin, A., J. Polym. Sci. Part B: Polym. Phys., 30 (1992) 1247. 4. Kimmich, R., Kopf, M. & Callaghan, P., J. Polym. Sci. Part B: Polym. Phys., 29 (1991) 1025. 5. Charlesby, A. & Jaroszkiewicz, E. M., Eur. Polym. J., 21 (1985) 55. 6. Brereton, M. G., Macromolecules, 22 (1989) 3667. 7. Brereton, M. G., Macromolecules., 23 (1990) 1119. 8. Brereton, M. G., Progress in Colloid and Polymer Science, 90 (1992) 96. 9. Brereton, M. G., Ward, I. M., Boden, N. & Wright, P., Macromolecules, 24 (1991) 2068. 10. Ries, M. E., Brereton, M. G., Cruickshank, J. M., Klein, P. G. & Ward, I. M., Macromolecules, 28 (1995) 3282. 11. Levitt, M. H. & Freeman, R., J. Magnetic Resonance, 43 (1981) 65. 12. Klein, P. G., Ladizesky, N. H. & Ward, I. M., J. Polym. Sci. Part B: Polym. Phys., 24 (1986) 1093. 13. Dounis, P. & Feast, W. J., Polymer, 37 (1996) 2547. 14. Cohen-Addad, J. P., J. Chem. Phys., 71 (1979) 3689. 15. Brereton, M. G., J. Chem. Phys., 94 (1991) 2136. 16. Zoepfl, F. J., Markovic, V. & Silverman, J., J. Polym. Sci. Part B: Polym. Chem., 22 (1984) 2017. 17. Woods, D. W. & Ward, I. M., Plastics, Rubber and Composites Processing and Applications, 18 (1992) 255. 18. Jones, R. A., Taylor, D. J. R., Stepto, R. F. T. & Ward, I. M., Polymer, 37 (1996) 3643
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19. Klein, P. G., Ladizesky, N. H. & Ward, I. M., Polymer, 28 (1987) 393. 20. Brereton, M. G. & Klein, P. G., Polymer, 29 (1988) 970. 21. Klein, P. G., Brereton, M. G., Rasburn, J. & Ward, I. M., Makromol. Chem., Macromol.
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Edwards, S. F. & Vilgis, T., Polymer, 27 (1986) 483. Wool, R. P., Macromolecules., 26 (1993) 1564. Ball, R. C. & McLeish, T. C. B., Macromolecules, 22 (1989) 1911. Adams, C. H., Hutchings, L. R., Klein, P. G., Richards, R. W. & McLeish, T. C. B., Macromolecules, 29 (1996) 5717.
26. Anderson, P. W. & Weiss, P. R., Review of Modern Physics, 25 (1953) 269. 27. Doi, M., Edwards, S. F., The Theory of Polymer Dynamics. Clarendon Press, Oxford, 1986.
APPENDIX To calculate
the transverse relaxation from a reptating chain of molar mass M the model outlined in Section 3.1 is employed, recall Fig. 1. The local level dynamics are taken to be fast Abrb << 1 and the NMR problem is resealed onto the reptating R vectors. Each of these vectors is formed by connecting together N, Rouse units, this being related to the average molar mass between entanglements Me.The transverse relaxation is then determined from eqn (4), which can be usefully rewritten in terms of the Cartesian components of the vector (R,,R,,R,) as G(t) = (exp[$$
f (2Rz - R; - R:)]) 0
(A.1)
with
(A4 Since the Cartesian reduced to
components
G(t) =
are independent,
the calculation
Re[s(2A,,t>g(-A,,t)g(-h,,t)l
can be
(A.3)
with
g@d) = where the problem now g(A,,t) for a reptating chain employed. The validity of where it was noted that this
(exp[ $$[ E])
(A-4
is the evaluation of g(AR,t). To calculate the second moments approximationz6 shall be this technique was examined by Brereton,” method was accurate until the chain’s motion
304
M. E. Ries et al.
became too highly restricted. In later work by Ries et ~1.“’ on linear poly(ethylene oxide) chains the approximation was found to be valid for chains greater than MC, i.e. in the entangled regime. This approximation makes use of the well-known result for a random Gaussian variable X
(expW)>= exp[Gf>+ 3CZ2)- WY)11 where (. . *) m ’ d’ica t es an average value found over a Gaussian In the transverse relaxation calculation (A.4) is compared From this likening the term X is given by
(A.5) distribution. with (AS).
(A.61 If the integral is replaced by a sum over discrete time and the dynamics are taken to be fast, then X becomes a summation of essentially independent random variables. This then produces the required Gaussian statistics. The term (X) is linear in AR and so will not survive the construction of G(t) from g(AR,t), see eqn (A.3). Applying eqn (AS) to the NMR problem and ignoring this linear term gives ((R;(t’)R;(t”))
- (R:(r’))(R;(t”)))dt’dt”
1 (A.7)
Since a Gaussian distribution is completely specified by its mean and variance the fourth-order correlation term presented above, (Rf(t’)R:(t”)), can usefully be rewritten from another standard result (R:(t’)R:(t”))
= (2R,(t’)R,(t”))*
+ (R;(t’))(R;(t”))
(A.8)
to give g(AR,f) = exp[ -( $)’
[[(R,(t’)R,(t”)Y
dr’dt”]
(A.9)
This result is the general starting point for a second moments approximation calculation. To proceed a bond correlation function is needed, in this instance that corresponding to entangled chains. For a polymer comprising N’ = M/M, reptating segments the bondbond correlation function for the R vectors is that of the primitive path for an entangled chain, given by27 (R,,j(t’)R7,j(t”))
=g
g sin’/%) P 0
exp(-
e)
(A.lO) I>
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Proton NMR investigation of polyethylene melts
with z’ep P
=
12N,’ Nt3b2q,
(AX)
z2a2p2
and
b21
(A.12)
” = 12kT
where l is the monomer friction coefficient, a is the mesh size, which will be taken to be X@$, and k the Boltzmann constant. Next the NMR bond is placed at the centre of the reptating chain j = N’/2, this simplifies eqn (A.lO) to (R,(t’)R,(t”))
=
$$‘:$:‘2 exp(
- v)
(A.13) 2p+l
The calculation now commences completed to give G(t)=exp
at (A.9). The integrals
p { -6 (AR) 2’~~~ [2(2,,,)‘( $+
exp(jZ)
over time are
- I)])
(A-14)
with 1 -=rep r P24 For fast reptation
1 feP 2ptl
+-
1 Gy+
(A.15) 1
dynamics (NzN’3rb -CCt), eqn (A.14) simplifies to G(t) = exp[ -I,,($
,“rzr
rp,,]
Next N’ is taken to be large and the summations This gives G(t) = exp[ -lFAzrb$ln(x)I e
replaced
e
(A.16) by integrals.
(A.17)
The molar mass signature of a reptating chain is, therefore, distinct from a Rouse chain. The exponent is predicted to have a stronger molar mass dependence than the Rouse model by a factor M.
ELSEVIER
A proton NMR investigation entanglements in polyethylene polymers M. E. Ries,” M. G. Brereton;
of crosslinks and networks and star
P. G. Klein”* & P. Dounis,bt
“IRC in Polymer Science and Technology University of Leeds, Leeds LS2 9JT, UK bIRC in Polymer Science and Technology, University of Durham, Durham DHl 3LE, UK (Received
2 August
1996; revised
17 December
1996; accepted
18 December
1996)
ABSTRACT This paper investigates the ability of proton transverse NMR relaxation to study the structure and dynamics of polyethylene melts. Commercial linear samples of similar M,, and various M,,, have been examined, together with samples crosslinked by electron beam irradiation. Apart from the linear sample of lowest M,, all the free induction decays (FIDs) are very similar and this is attributed to the presence of free chain ends. The magnitude of this problem is illustrated by generating theoretical FIDs from networks of different crosslink density and reptating linear chains of different entanglement spacings, with and without free chain ends. A new result for the FID using the reptation spectrum is presented and used to fit data from the linear polyethylenes. This yields an “NMR average” molar mass, similar to the accepted value and provides information on a sample’s polydispersity. Three-arm polyethylene star polymers, of low polydispersity and various arm lengths, have also been studied. The large variation in the chain dynamics from the core to the free end of the star arm is not reflected in the FIDs, which can all be well-fitted by double exponentials. This highlights the importance of specific isotopic labelling in studying such systems. 0 1997 Elsevier Science Limited
1 INTRODUCTION Commercial polyethylene samples contain protons at all locations within the material. A transverse proton NMR measurement thus records the * Author
to whom correspondence should be addressed. t Present address: Exxon Chemicals Ltd, PO Box 1, Milton OX13 6BB, UK.
Hill, Abingdon,
Oxfordshire
286
M. E. Ries et al.
cumulative signal from the different locations and presents an ensemble average. This complicates the interpretation and sets limits on what can be uncovered by proton NMR. In this work a range of commercial samples and specially synthesised low polydispersity star polymers are investigated to ascertain the extent to which proton NMR can reveal melt structure and dynamics without the use of specific labelling. The analysis of small linear chains undergoing fast dynamics is relatively simple, since the proton transverse free induction decay mainly displays a single exponential.’ However, many commercial polymeric systems can generate complex non-exponential decays.2 One of the main problems encountered in NMR analysis of polymer melts has been interpreting these signals in terms of structural and dynamical properties. Until recently these decays have been described by empirical formulae, with much effort being made to attribute the phenomenological terms to molecular correlation times, entanglement densities, crosslink densities and gel fractions. ‘-5 A framework to interpret the proton NMR transverse decay for a polymer chain in a variety of systems, ranging from short Rouse chains’ to been introduced into the entangled melts’ and networks,8 has recently literature. These techniques have already been employed to investigate monodisperse linear chains, including poly(ethylene)’ and poly(ethylene oxide).‘” In this work the investigation is extended from these relatively ideal systems to a range of complex structures; including highly polydisperse entangled linear melts, networks formed by electron irradiation and low polydispersity star polymers. The networks formed from the electron irradiation of poly(ethylene) melts illustrate the difficulty in distinguishing between crosslinks and entanglements by proton NMR. The abundance of free pendant chains and other mobile constituents formed by main chain scission mask the information in the transverse relaxation function concerning entanglement and crosslink density. The signal from a model theoretical network with and without free ends is then used to illustrate the potential of NMR to measure crosslink density and the problem introduced by “rogue” free ends. A new result for the free induction decay calculated from the reptation spectrum is presented and employed to interpret the linear entangled data. This gauges how sensitive proton NMR is to a sample’s polydispersity. A theoretical model is then used to demonstrate the problem induced by free ends. These peripheral segments of the reptating chain undergo fast reorientation and with this comes a gradual decay in the transverse relaxation. This again hides the detail concerning the more rapid NMR decay from the central reptating segments.
Proton NMR investigation of polyethylene melts
287
Star polymers constructed at the University of Durham, which form the precursor to model networks, are also investigated. These offer a wide distribution of correlation times, from Rouse-like ends to highly constrained reorientation at the star’s core. These, thus, indicate the sensitivity of proton NMR to a specific spectrum of molecular relaxation times. This paper thus points out the strengths and weaknesses of proton NMR and sets out an analytic approach to interpreting the transverse relaxation from polymeric samples.
2 EXPERIMENTAL 2.1 NMR measurements All the NMR measurements were made on a Chemagnetics CMX-200 spectrometer, operating at 200 MHz for protons. The Levitt-Freeman” modification of the Carr-Purcell echo train was employed:
This self-compensating sequence ensures that every second echo is correct, providing the diffusional processes are not significant. Full quadrature phase cycling was employed, with the (7c/2) pulse lasting 2 ps and the dephasing time T typically l/2 ms. A total of n/2 = 200 points was acquired, at the peak of every second echo, with the resultant free induction decay being the average of 64 accumulations. 2.2 Sample details Three samples consisted of linear polydisperse chains with a similar number average molar mass, M, = 25 000 g/mol, but with different weight average molar mass M,,,. Two samples were spun poly(ethylene) monofilaments, which had been electron-irradiated at room temperature in a vacuum, one receiving a dose of 6.0Mrad and the other 3.5 Mrad. The details of sample preparation can be found in a previous paper.‘* The above information is summarised in Table 1. Three-armed polyethylene stars were produced by quenching living polypentenamer with 1,3,5benzenetrialdehyde and hydrogenating.13 The
288
M. E. Ries et al.
Linear Name
and electron
M, (before
AL7050 UN2912 HO20 P2 P5
branch lengths Table 2.
TABLE 1 beam crosslinked
treatment)
polyethylene
M, (before treatment)
glmol
glmol
24 24 26 28 28
63 000 221000 450 000 115000 115 000
000 000 000 000 000
were varied
and the three
samples
samples Treatment
none none none 3.5 Mrad in vat 6.0 Mrad in vat
investigated
are listed in
3 THEORY 3.1 Transverse
NMR relaxation
Dipolar interactions of the proton pairs, which are attached to the polymer backbone, cause the dephasing of the transverse nuclear magnetic components. Experimentally, the relaxation of the transverse magnetisation is recorded as the free induction decay G(t). This function is sensitive to both the correlation times of the various chain segments and the melt structure. A scale invariant model of the polymer chain for the NMR properties was first envisaged by Cohen-Addad14 and later developed by Brereton:” several monomers N, are connected together to form a statistical unit b with the dipolar NMR interaction averaged over this length scale. The
The
star
Name star 1 star 2 star 3
TABLE 2 polymer molar polydispersities
mass
M, (g/m4
MJM.,
49 800 62 800 95 300
1.42 1.54 1.54
and
Proton NMR investigation of polyethylene melts
289
Fig. 1. The hierarchical model for the NMR properties of a polymer chain governed by two distinct modes of relaxation. N, monomers are connected together to form one b unit, with a further Nh b segments forming a vector R.
transverse
relaxation function G(t) for the chain is then specified by15 G(t) = (co, [ +
I’ (3 co? &‘,- l)]) 0
where 0, is the angle between the statistical unit b containing the NMR probe and the magnetic field (see Fig. l), Ab is the resealed dipolar interaction constant corresponding to the statistical length b given by
with y the gyromagnetic ratio for a proton, d the proton-proton distance within the proton pair and p0 the permeability of a vacuum. The (. * -) in eqn (1) indicates an averaging of the NMR active statistical segment over all available conformations, in the time interval 0 to t, subject to any constraint that the environment imposes on the bond, e.g. crosslinks or entanglements. It is through this averaging that the NMR probe is sensitive to the polymer chain environment. The problem posed by the evaluation of eqn (1) has been treated in a series of papers.C8,‘5 For completeness the main results will be summarised in the appropriate sections.
M. E. Ries et al.
290
3.2 Scale invariance Entangled polymer chains in a melt form a hierarchical structure of dynamics, with Rouse dynamics occurring on a small scale and reptation on a larger one. We have shown that when the local dynamics are fast A,z, << 1, where z, is the correlation time of bond b, it is then possible to rescale further7 the NMR interaction onto the slower, larger scale motion of the chain. For example, consider the chain as a series of vectors R each formed by connecting Nb statistical units b together as illustrated in Fig. 1. In the regime of fast local level dynamics the transverse relaxation no longer depends on the rate of reorientation of these statistical units, but only on the motion of the larger scale segments R. This approximation is useful since it renders the precise form of the local dynamics unimportant, simplifying the NMR calculation and allowing a coarser grained view of the polymer molecule. The dipolar constant rescales further to AR, where7
The transverse
relaxation G(t) =
where
is then determined
(cos[F I’(3 cos2OR- l)]) 0
8, is the angle the vector
4 RESULTS 4.1 Poly(ethylene)
from
entangled
R makes
AND linear
with the magnetic
(4) field.
DISCUSSION
chains
and networks
Much research has gone into investigating the effects of electron beam properties of structure and mechanical irradiation on the This bombardment of high energy electrons results poly(ethylene).12.1”2’ in crosslinking and chain scission, creating a complex micro-structure. The final network configuration might contain pendant chains, trapped entanglements and closed loops. Ideally, one would like to know how all these influence the macroscopic properties of the sample. A first step towards this goal would be the determination of the melt micro-structure. Work2’,” carried out at the University of Leeds on poly(ethylene) networks using the Edwards-Vilgis2* “ Slip Link Model” to interpret force
Proton NMR investigation of polyethylene melts
291
extension data revealed that entanglements were quite loose and, therefore, distinct from permanent chemical crosslinks. The question is whether this difference would be revealed in an NMR experiment. A theoretical model will, thus, be employed to investigate the ability of NMR to distinguish the dephasing of protons held between entanglement points from those connected by crosslinks. The main objective of this work is to examine the use of proton NMR to analyse network structures. To this aim several linear poly(ethylene) samples of similar number average molar mass are investigated using transverse NMR. Two samples have been subjected to different amounts of electron irradiation with a further three being untreated, but having various polydispersities. This allows the direct comparison between networks with expected distinct crosslink densities and those with the linear entangled chains, to give an insight into whether NMR can distinguish between permanent and temporary crosslinks. Measurements of the transverse relaxation on the polydisperse untreated samples provide access to whether NMR can be employed to determine molecular weight (m.w.) distributions. It will be theoretically shown that the transverse relaxation function for a reptating chain is expected to display an approximately linear m.w. dependence as opposed to the weaker log(m.w.) found’,” for a Rouse chain. Whether this is sufficient to reveal the distribution of chain sizes in a poly(ethylene) melt will be indicated by the NMR measurements on the untreated samples. This work builds on a previous paper’ by Brereton et al., where well-characterised linear poly(ethylene) chains were investigated. From measurements of the transverse magnetisation and the employment of recently developed theories it was possible to observe both Rouse and reptation dynamics. This work also determined for poly(ethylene) the proton dipole coupling constant A6 = 5606s’, the fundamental Rouse relaxation time r, = 1.49 X lo-‘s (at 149°C) and the size of a Rouse unit 252 g/mol. These parameters found from the analysis of monodisperse linear chains are essential to the investigation of uncharacterised melts. The transverse decays from the poly(ethylene) samples, which show a range of polydispersity and network fractions, corresponding to Table 1 are displayed in Fig. 2. The first thing to notice is the similarity between the results. Apart from the AL7050 sample, which has the lowest polydispersity, the data lies on approximately the same curve. There are slight differences in the long time data. The transverse relaxations fan out, with the more rigid melts decaying slightly faster. For example material P5, which underwent the most electron irradiation and has a significant gel fraction ofl’ 0.55, is below the others in Fig. 2, with the linear melt with the greatest polydispersity being next.
M. E. Ries et al.
I
z
P5 AL7050
-3
-4
time/s
Fig. 2.
Transverse NMR relaxation of various poly(ethylene) lines through the data are fits to bi-exponential
samples, see Table functions.
1. The
Apart from this minor distinction an entangled melt of linear chains is essentially indistinguishable, as far as NMR is concerned, from a network. Indeed it is possible to fit the data quite adequately using a phenomenological decay Gbi-exp(t), see Fig. 2, Gbi_,,,(t) = a exp(-at)
+ (1 - a>exp(-@)
(5)
where a, cz and p are fitting constants. This reveals that the various forms of relaxation undergone by the different components within the melt become translated by the NMR experiment into a rigid and flexible component. Information about polydispersity, crosslink/entanglement density and chain scission has been lost in the resultant signal, as all the decays are similar. A major restriction on the ability of proton NMR to decipher melt configuration is the abundance of free ends in a melt. These dangling ends, or even small segments that have been cut from the main chain by electron bombardment, are highly mobile. Fast reorientation, as anticipated with these components, would generate a slowly decaying signal. Without labelling, i.e. the specific placing of NMR active nuclei at known locations, NMR measures the summated signal from all the protons in the
293
Proton NMR investigation of polyethylene melts
sample. This essentially averages the decay over the various motions undergone in the melt. The result is that any slow reorientating chains with correspondingly fast decaying signals are hidden by the more gradual decay of the mobile constituents. The information of interest, corresponding to reorientation restricted either by entanglements or chemical crosslinks, is lost in favour of these freer chains. The magnitude of this problem will be illustrated in the following two sections. 4.2 Theoretical
FIDs from crosslinked
and entanglement
networks
The more mobile free ends significantly reduce the potential of NMR to investigate uncharacterised melts. This can be highlighted by a theoretical example in which the signals from three model networks, of different crosslink densities, are contrasted with those from the same networks containing free ends. Each network is formed by crosslinking 10 segments of equal molar mass, M,, together as shown in Fig. 3(a). To disturb the model network chain, by the inclusion of free ends, the peripheral crosslinks are released as in Fig. 3(b). To generate the theoretical decay for the model networks it is assumed that the crosslink points are essentially fixed in space. This represents the effect a network would have on its constituent chains. A chain in the network is unable to diffuse far from its starting point without becoming hindered by the junction points. The chain is now pictured as a series of vectors R, recall Fig. 1, that link successive junction points together. The relaxation time for this vector is very large in comparison with the time scale of the NMR experiment. In this regime, known as “frozen” bond
Fig. 3.
Showing:
(a) network
chain
with no free ends; ends.
(b) network
chain with two free
294
M. E. Ries et al.
dynamics, the signal becomes insensitive to the time. The solution to eqn (4) in this regime isl’ Gmodel (t) = Re{(l - 2iA,t)-“2(1
vector
R's correlation
(6)
+ iA,t))l>
network
where AR is the resealed interaction strength crosslink density through (compare with eqn (3))
and iV, is the number
of Rouse
units between
corresponding
crosslink
points
to
the
given by
with mousebeing
the molar mass of a Rouse unit = 252 g/mol for poly(ethylene). Work by Kimmich et ~1.~ revealed that the outer entanglement lengths of a reptating chain behaved, as far as NMR is concerned, in a similar fashion to a Rouse chain. This was later confirmed by Ries et d.” in studies of poly(ethylene oxide) linear chains. Thus, for the case of the network with free ends the signal generated from these mobile segments can be approximated by that expected from a Rouse chain of length N,. The above signal becomes modified to G TT,2Lidr(t) = 2 Re((1 - ZA,t))“*(l
+ i&t>-‘1
+ 4 exp
-6A2,r,
ln(N,) t n
where r, is the fundamental Rouse relaxation time, which for polyethylene at 149°C is 1.49 X lo-’ s.~ The second term in the above equation corresponds to the expected signal from a Rouse chain as predicted and measured in work by Brereton and co-workers9~“’ Contrasting the signals generated by the above equations, Fig. 4, reveals how the information about M, is concealed within the chain end decay. The information corresponding to the crosslink density is found in the long time data. The inclusion of more mobile free ends, with their correspondingly slow decaying signals, covers these details. Information is still seen about the crosslink density in the signal from the “imperfect” network, but in comparison to the perfect network it is greatly reduced. To observe motion corresponding to that between crosslink points the chains need to be selectively labelled so as to remove the influence of the rogue free ends. We will now investigate the theoretical FID from a reptating chain in a
295
Proton NMR investigation of polyethylene melts time/s 0.02
Fig. 4.
Theoretical
transverse
0.03
0.04
0.05
relaxations from networks of different crosslink densities, with and without free ends.
melt. The transverse decay from a 25 OOOg/mol chain with labelled peripheral segments will be compared to one with unlabelled (i.e. contains no nuclei that contribute to the NMR decay curve) outermost segments, see Fig. 5(a, b). To achieve this a theoretical decay from a reptating melt is required, this is derived in the appendix. In the regime of fast large scale reorientation
(5
r, << t ) the relaxation is specified by
a)
Fig. 5.
Showing: (a) entangled
chain; (b) entangled
chain with unlabelled
ends.
M. E. Ries et al.
296
time/s 0
Fig. 6.
Theoretical
0.02
0.01
0.03
0.04
0.05
transverse relaxation functions for reptating 25 000 g/mol unlabelled ends for different entanglement densities.
The expression (10) generates the unlabelled ends. To include these modified as before, see eqn (9), to
decay for a reptating the above relaxation
G,,ptation (t) = M -M2M’exp[ -tF tfrre ends f-
chains,
with
chain with function is
Azr,gln(g)] C
P
2M, M
The signal generated by the entangled chain experiencing different entanglement densities Me (centred on the M, for poly(ethylene) of 2000 g/mol)‘” with unlabelled ends is shown in Fig. 6. The effect of free ends can then be compared in Fig. 7. The entanglement molecular weight dependence of the reptation spectrum survives the inclusion of Rouse-like free ends. In the unirradiated samples the entanglement densities are, of course, the same, the value being a characteristic of poly(ethylene), and this explains the similarities of the data, recall Fig. 2. 4.3 Crosslinks
vs entanglements
It is clear from the above two theoretical analyses that entanglements are ostensibly distinct from crosslink points in an NMR experiment, compare Figs 4 and 6. This will be true until the motion between the entanglement points becomes so sluggish that it appears frozen on the NMR time scale. Then the reptating signal will be identical to a network with a crosslink density equal to the entanglement density.
297
Proton NMR investigation of polyethylene melts time/s
-4 -5
i
-6 I
Fig. 7.
Theoretical transverse relaxation functions for reptating 25 000 g/mol different entanglement densities together with free ends.
chains
for
Inclusion of any free chains or imperfections would reduce the maximum molar mass of a reptating polymer that would be discernible from a crosslinked network. In Fig. 8 the signal from a 100000 g/mol reptating chain with unlabelled ends is compared with that from a model network. For a correct comparison the crosslink density has been set to the poly(ethylene) entanglement density of (=2000g/mol). The two transverse relaxations follow each other until they have decayed to about 5% (eP3) of their starting values. It is, thus, anticipated that these slight differences will not be discernible for a less model system.
time/s 0.02
Fig. 8. Theoretical transverse relaxation from 2OOOg/mol compared with a reptating chain 100 000 g/mol and entanglement
0.03
a model network of crosslink density with unlabelled ends of molar mass density 2000 g/mol.
M. E. Rim et al.
298
4.4 Reptation
fit to the linear un-irradiated
poly(ethylene)
samples
We now assess the predicted reptation relaxation on the untreated poly(ethylene) samples. The first experimental result to be investigated is the sample AL7050, which has the lowest polydispersity. The data will be fitted to an equation similar in form to (11) as
The value of Me is set at the entanglement density of poly(ethylene) =2000 g/mol. The free ends have been given a transverse relaxation for a Rouse chain of an entanglement length, as suggested by Kimmich et u!.,~ but modified by a factor cz. The ends are expected to reorientate slower than that of free chains, simply because they are tethered to a reptating polymer. The parameter (Yallows for this and gives an insight as to how distinct the free ends are to Rouse chains. It is thus anticipated that LY2 1. From the free fit, see Fig. 9: (Y= 2.89, which is of order unity as 1.0
0.8
0.6
0.2
0.1
Fig. 9.
Reptation
0.2
time/s fits to linear chain data.
0.3
Proton NMR investigation of polyethylene melts
299
expected and M = 19 400 g/mol with this being close to the number average molar mass of the sample 24000 g/mol. The closeness of the fit and the reasonable nature of the parameters supports the use of the reptation spectrum. It is interesting to ask what effective molar mass is required to fit the other linear samples. The difference between the AL7050 material and the others is essentially the molar mass distribution. The above values can thus be used, only leaving one free parameter M. The HO20 sample is fitted to the above expression (12) in Fig. 9, but with the previous parameters, other than M, being employed. This gives a value for M of 27 000 g/mol. Again the fit is reasonable and as expected M is slightly larger than the number average molar mass expressing the wider molecular weight distribution. 4.5 Polyethylene
stars
As part of the work being carried out in the IRC at the University of Leeds towards the ability of NMR to interpret network dynamics several poly(ethylene) stars have been made in the synthesis group at the University of Durham. The stars used in this analysis form the precursors to model networks. Stars exhibit markedly different reorientation to that of linear chains. They are unable to reptate due to the core of the star acting like an anchor in the melt of stars. A theory put forward by Ball and McLeish proposes that the star reorientates by “breathing modes”,24 the end of an arm has to retract along its length before protruding out into its surrounding matrix to adopt a new random conformation. For the stress on segments deep within the star (towards the core) to relax, it requires the complete contraction of the arm. This event is highly unlikely with the result being that the core is essentially “frozen”. The star polymers offer a unique distribution of correlation times, ranging from Rouse-like for the peripheral segments up to locked in orientations at the core. This presents an opportunity to see how capable NMR is in differentiating between various mechanisms of reorientation. These results can then be directly contrasted with the signal from linear chains of comparable molar mass to a star arm. This indicates the ability of proton NMR to distinguish different relaxation mechanisms and to analyse uncharacterised melts. The signal from the poly(ethylene) stars can be found in Fig. 10. Stars by nature of their junction point have fewer free ends per chain (arm). This coupled with the anticipated spectrum of relaxation generated by the tethering of the core might be expected to give rise to a characteristically different decay. However, this is not so. As with the poly(ethylene)
M. E. Ries et al.
300 0
-1
2 13 ,e
-2
-3
-4 U
time/s Fig. 10.
Transverse NMR relaxation of various poly(ethylene) stars and the linear HO20 sample. The lines through the data are fits to bi-exponential functions.
network results the data is reliably fitted by a bi-exponential function identical in form to eqn (5), see Fig. 10. Again the summation of the signal over all the proton pairs, constrained to mobile, results in a two component fit. Instead of being able to decipher the nature of the spectrum of relaxations undergone by the star, it is possible to visualise the star as equally rigid inner segments with a flexible free end. This loss of information is highlighted when star 1 is compared with a linear chain of similar number average molar mass to its arm in Fig. 10. This shows the lack of distinguishing features present in the signal for the star. If it is nearly impossible to differentiate between star polymers and linear chains using proton NMR then it cannot be expected to characterise unknown melt micro-structures. This again reminds us of the importance of labelling if information is to be learnt from transverse NMR measurements. 5 CONCLUSIONS In this paper the attention was focused interpret structure and dynamics from
on the ability of proton NMR to commercial polymer melts. This
Proton NMR investigation of polyethylene melts
301
possibility was investigated both theoretically and experimentally. Several various linear polyethylene samples, some of which had undergone intensities of electron beam irradiation, were observed using the transverse relaxation decays. This highlighted the problems encountered when examining unknown melt micro-structures. Previous force extension analysis performed on the same samples had revealed information about entanglement and crosslink densities and even the freedom of entanglements to slide along the chain.” However, NMR found it difficult to even distinguish between these samples. It was proposed that the different environments, reptating melt or networked chains, were concealed by the presence of dangling chain ends. A model was constructed to illustrate the problem imposed by Rouse-like free ends in the network. Two networks were compared, both of these had a well-defined molar mass between crosslinks, but one included dangling peripheral sections. At long times, Fig. 4, it was possible to distinguish crosslink densities for the perfect network. The inclusion of the freer more mobile end sections generated a slowly decaying signal that reduces the discernible information about crosslink density, recall Fig. 4. The untreated polyethylene samples had various polydispersities. The reptation time depends strongly on the molar mass of a chain. It was expected that these different distributions of chain length would be revealed through the decay. However, the transverse relaxations from these samples, apart from the sample of lowest M,, were very similar. Also, the signals could be well-fitted by a bi-exponential, indicating the lack of information revealed in the decay. The reptation spectrum was employed to calculate the expected signal from an entangled melt. This was then used to fit the signal from the linear chains. From this it was possible to measure an NMR average molar mass of the chain, which compared well to the accepted value. To weigh the ability of NMR to distinguish crosslinks from entanglements the theoretical signals from a network and a reptating melt were compared. The crosslink density was set to the entanglement density. This then allowed direct comparison between the influence of entanglements and crosslink points. It was clear that until the reptating chain became too large, lO’g/mol for poly(ethylene) at 14972, the motion of protons between entanglements is distinguishable from that between crosslinks by NMR. As part of preliminary work on model networks, a selection of polyethylene stars have been studied. These stars exhibit markedly different molecular relaxation to linear chains, stemming from the anchored core that disallows reptation. Rheological measurementsz5 have verified the theoretically predicted “breathing modes”, with this technique
M. E. Ries et al.
302
being readily able to distinguish the stars from linear chains. The NMR transverse decays for the polyethylene stars were bi-exponential in nature, recall Fig. 10. This makes it impossible to assign or confirm any form for the spectrum of relaxation modes. A linear chain with molar mass comparable to the arm of a star was experimentally contrasted with these results. Here it was found that the two signals were similar, highlighting the weakness of proton transverse NMR to investigate polymer melts without the use of specific labelling. We are currently pursuing this, studying selectively deuterated polybutadiene star polymers.
ACKNOWLEDGEMENT We wish to thank manuscript.
Professor
W. J. Feast
for his helpful
comments
on the
REFERENCES 1. Charlesby, A., Radiat. Phys. Chern., 39 (1992) 45. 2. Cohen-Addad, J. P. & Dupeyre, R., Polymer, 24 (1983) 400. 3. Bremner, T. & Rudin, A., J. Polym. Sci. Part B: Polym. Phys., 30 (1992) 1247. 4. Kimmich, R., Kopf, M. & Callaghan, P., J. Polym. Sci. Part B: Polym. Phys., 29 (1991) 1025. 5. Charlesby, A. & Jaroszkiewicz, E. M., Eur. Polym. J., 21 (1985) 55. 6. Brereton, M. G., Macromolecules, 22 (1989) 3667. 7. Brereton, M. G., Macromolecules., 23 (1990) 1119. 8. Brereton, M. G., Progress in Colloid and Polymer Science, 90 (1992) 96. 9. Brereton, M. G., Ward, I. M., Boden, N. & Wright, P., Macromolecules, 24 (1991) 2068. 10. Ries, M. E., Brereton, M. G., Cruickshank, J. M., Klein, P. G. & Ward, I. M., Macromolecules, 28 (1995) 3282. 11. Levitt, M. H. & Freeman, R., J. Magnetic Resonance, 43 (1981) 65. 12. Klein, P. G., Ladizesky, N. H. & Ward, I. M., J. Polym. Sci. Part B: Polym. Phys., 24 (1986) 1093. 13. Dounis, P. & Feast, W. J., Polymer, 37 (1996) 2547. 14. Cohen-Addad, J. P., J. Chem. Phys., 71 (1979) 3689. 15. Brereton, M. G., J. Chem. Phys., 94 (1991) 2136. 16. Zoepfl, F. J., Markovic, V. & Silverman, J., J. Polym. Sci. Part B: Polym. Chem., 22 (1984) 2017. 17. Woods, D. W. & Ward, I. M., Plastics, Rubber and Composites Processing and Applications, 18 (1992) 255. 18. Jones, R. A., Taylor, D. J. R., Stepto, R. F. T. & Ward, I. M., Polymer, 37 (1996) 3643
Proton NMR investigation of polyethylene melts
303
19. Klein, P. G., Ladizesky, N. H. & Ward, I. M., Polymer, 28 (1987) 393. 20. Brereton, M. G. & Klein, P. G., Polymer, 29 (1988) 970. 21. Klein, P. G., Brereton, M. G., Rasburn, J. & Ward, I. M., Makromol. Chem., Macromol.
22. 23. 24. 25.
Symp., 30 (1989) 45.
Edwards, S. F. & Vilgis, T., Polymer, 27 (1986) 483. Wool, R. P., Macromolecules., 26 (1993) 1564. Ball, R. C. & McLeish, T. C. B., Macromolecules, 22 (1989) 1911. Adams, C. H., Hutchings, L. R., Klein, P. G., Richards, R. W. & McLeish, T. C. B., Macromolecules, 29 (1996) 5717.
26. Anderson, P. W. & Weiss, P. R., Review of Modern Physics, 25 (1953) 269. 27. Doi, M., Edwards, S. F., The Theory of Polymer Dynamics. Clarendon Press, Oxford, 1986.
APPENDIX To calculate
the transverse relaxation from a reptating chain of molar mass M the model outlined in Section 3.1 is employed, recall Fig. 1. The local level dynamics are taken to be fast Abrb << 1 and the NMR problem is resealed onto the reptating R vectors. Each of these vectors is formed by connecting together N, Rouse units, this being related to the average molar mass between entanglements Me.The transverse relaxation is then determined from eqn (4), which can be usefully rewritten in terms of the Cartesian components of the vector (R,,R,,R,) as G(t) = (exp[$$
f (2Rz - R; - R:)]) 0
(A.1)
with
(A4 Since the Cartesian reduced to
components
G(t) =
are independent,
the calculation
Re[s(2A,,t>g(-A,,t)g(-h,,t)l
can be
(A.3)
with
g@d) = where the problem now g(A,,t) for a reptating chain employed. The validity of where it was noted that this
(exp[ $$[ E])
(A-4
is the evaluation of g(AR,t). To calculate the second moments approximationz6 shall be this technique was examined by Brereton,” method was accurate until the chain’s motion
304
M. E. Ries et al.
became too highly restricted. In later work by Ries et ~1.“’ on linear poly(ethylene oxide) chains the approximation was found to be valid for chains greater than MC, i.e. in the entangled regime. This approximation makes use of the well-known result for a random Gaussian variable X
(expW)>= exp[Gf>+ 3CZ2)- WY)11 where (. . *) m ’ d’ica t es an average value found over a Gaussian In the transverse relaxation calculation (A.4) is compared From this likening the term X is given by
(A.5) distribution. with (AS).
(A.61 If the integral is replaced by a sum over discrete time and the dynamics are taken to be fast, then X becomes a summation of essentially independent random variables. This then produces the required Gaussian statistics. The term (X) is linear in AR and so will not survive the construction of G(t) from g(AR,t), see eqn (A.3). Applying eqn (AS) to the NMR problem and ignoring this linear term gives ((R;(t’)R;(t”))
- (R:(r’))(R;(t”)))dt’dt”
1 (A.7)
Since a Gaussian distribution is completely specified by its mean and variance the fourth-order correlation term presented above, (Rf(t’)R:(t”)), can usefully be rewritten from another standard result (R:(t’)R:(t”))
= (2R,(t’)R,(t”))*
+ (R;(t’))(R;(t”))
(A.8)
to give g(AR,f) = exp[ -( $)’
[[(R,(t’)R,(t”)Y
dr’dt”]
(A.9)
This result is the general starting point for a second moments approximation calculation. To proceed a bond correlation function is needed, in this instance that corresponding to entangled chains. For a polymer comprising N’ = M/M, reptating segments the bondbond correlation function for the R vectors is that of the primitive path for an entangled chain, given by27 (R,,j(t’)R7,j(t”))
=g
g sin’/%) P 0
exp(-
e)
(A.lO) I>
305
Proton NMR investigation of polyethylene melts
with z’ep P
=
12N,’ Nt3b2q,
(AX)
z2a2p2
and
b21
(A.12)
” = 12kT
where l is the monomer friction coefficient, a is the mesh size, which will be taken to be X@$, and k the Boltzmann constant. Next the NMR bond is placed at the centre of the reptating chain j = N’/2, this simplifies eqn (A.lO) to (R,(t’)R,(t”))
=
$$‘:$:‘2 exp(
- v)
(A.13) 2p+l
The calculation now commences completed to give G(t)=exp
at (A.9). The integrals
p { -6 (AR) 2’~~~ [2(2,,,)‘( $+
exp(jZ)
over time are
- I)])
(A-14)
with 1 -=rep r P24 For fast reptation
1 feP 2ptl
+-
1 Gy+
(A.15) 1
dynamics (NzN’3rb -CCt), eqn (A.14) simplifies to G(t) = exp[ -I,,($
,“rzr
rp,,]
Next N’ is taken to be large and the summations This gives G(t) = exp[ -lFAzrb$ln(x)I e
replaced
e
(A.16) by integrals.
(A.17)
The molar mass signature of a reptating chain is, therefore, distinct from a Rouse chain. The exponent is predicted to have a stronger molar mass dependence than the Rouse model by a factor M.