We present a consistent explanation for the appearance of “lozenge” shapes in contour plots of the two dimensional scattering intensity from stretched polymer networks. By explicitly averaging over quenched variables in a tube model, we show that lozenge patterns arise as a result of chain material that is not directly deformed by the stretch. We obtain excellent agreement with experimental data

A. Ramzi

C. Hayes

C. Rouf

D. J. Read

E. Straube

F. Boué

G. Heinrich

J. des Cloizeaux

M. Warner

R. C. Ball

R. T. Deam

S. F. Edwards

S. V. Panyukov

S. Westermann

T. C. B. McLeish

English

Read, D.J.

McLeish, T.C.B.

White Rose Research Online

This is a repository copy of ‘‘Lozenge’’ contour plots in scattering from polymer networks.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1502/Article:Read, D.J. and McLeish, T.C.B. (1997) ‘‘Lozenge’’ contour plots in scattering from polymer networks. Physical Review Letters, 79 (1). pp. 87-90. ISSN 0031-9007 https://doi.org/10.1103/PhysRevLett.79.87eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. VOLUME 79, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 7 JULY 1997‘‘Lozenge’’ Contour Plots in Scattering from Polymer NetworksD. J. Read and T. C. B. McLeishDepartment of Physics, University of Leeds, Leeds, LS2 9JT, United Kingdom(Received 9 December 1996)We present a consistent explanation for the appearance of “lozenge” shapes in contour plots of thetwo dimensional scattering intensity from stretched polymer networks. By explicitly averaging overquenched variables in a tube model, we show that lozenge patterns arise as a result of chain materialthat is not directly deformed by the stretch. We obtain excellent agreement with experimental data.[S0031-9007(97)03453-4]PACS numbers: 61.41.+e, 61.12.Bt, 83.80.DrRecently there have been a number of neutron scat-tering experiments on selectively-labeled and uniaxiallydeformed polymer networks and melts. A common ob-servation in these experiments is that contour plots of thetwo dimensional scattering intensity take on a diamond or“lozenge” shape. The experiments fall mainly into twocategories; for pure networks with dilute labeled chains[1] the lozenge pattern is permanent and does not relax,but if the labeled species is mobile the lozenge patternrelaxes and is a precursor to “butterfly” contour patterns[2,3]. In this Letter we demonstrate that the lozenge shapecan largely be explained by taking into account those chainsections in the system which are not directly deformed bythe stretch. We concentrate mainly on the case of pure net-works containing labeled chains, and discuss briefly howthe concept may be transferred to systems with a mobilespecies.In a polymer network, the effect of crosslinks andtopological entanglements is to localize the networkchains in space. Each monomer is confined to a regionaround its mean position and deviations from this resultin an elastic energy penalty due to the deformation ofthe surrounding network. To model this effect, Edwardsand co-workers [4] introduced the concept of a “tube”by placing each monomer on the chain in an isotropicharmonic localizing potential. The total free energyfunctional for the chain (in units ofkBT ) for a fixedconfiguration of potentials is thenFRhrlj ­12XlΩ3b2µ≠rl≠l∂212b2d4srl 2 Rld2æ, (1)where rl is the position of monomerl. The first termrepresents the Gaussian chain statistics (b is the steplength) and the second term represents the localizingpotentials, centered onRl (d is the tube diameter).The tube configuration given by theRl is “quenched”in the sense that it is fixed at the time of formationof the network. A stretch may be imposed by affinetransformation of theRl.This model was used by Warner and Edwards [5] topredict the neutron scattering function for a stretchednetwork in which a fraction of the chains is entirelydeuterated. They used the “replica trick” to perform theaverage over the quenched variablesRl. In terms of thestretch ratiolm in the three main axessm ­ x, y, zd theyfound a scattering function;Ssqd ­Z 10dxZ 10dx0 expΩ2XmQ2ml2mjx 2 x0j1 Q2ms1 2 l2mdd22p6R2g3∑1 2 expµ22p6R2gjx 2 x0jd2∂∏æ,(2)where Rg is the radius of gyration of the undeformedchain, Qm ­ qmRg is the normalized wave vector, andx ­ lyN andx0 ­ l0yN are chain contour length coordi-nates. Physically the first term in the exponential is thatobtained by affine deformation of a Gaussian chain. Thesecond term allows for local fluctuations about the meanpath of the chain.A modified form of the Warner-Edwards result hasbeen used recently by Straubeet al. [1,6,7] to interpreta series of experiments on stretched, labeled networks.One experiment [1] involved scattering from networksformed by randomly crosslinking melts which containeda small fraction of deuterated chains, providing a crucialtest for the Warner-Edwards result. It was found that theoriginal Warner-Edwards expression did not consistentlyfit the neutron scattering patterns obtained. Specifically,the expression does not yield the experimentally observedlozenges.Straubeet al. [1] presented modifications, empiricallyintroduced at the level of the final result of Warner andEdwards, which fitted the data well and were intended todescribe a deformation of the localizing tube-potential [8].However, we have found [9] that if the assumptions ofStraubeet al. are implemented at the fundamental levelof the model, it is impossible to derive their suggestedformula. Harmonic localizing potentials can give only aresult which is separable in the three main axes, yet theirsuggested formula does not have axis separability. Wefind that their assumptions cannot account for the lozengepattern.Other calculations making the approximations of phan-tom networks [10] or uniform density fields [11] have0031-9007y97y79(1)y87(4)$10.00 © 1997 The American Physical Society 87VOLUME 79, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 7 JULY 1997given lozenge scattering patterns, but have not been ableto identify their physical origin.Instead, we propose a physical mechanism for theappearance of lozenges in neutron scattering patterns fromdeformed networks. We suggest that in all the experi-ments which give lozenges, there exists chain materialwhich is not directly deformed by the strain. It is thecombination of deformed and undeformed material andthe interactions between them which gives rise to thelozenges. We illustrate this argument by taking a simplemodel for a pure network in which some of the chainpaths are labeled. Each labeled chain is constrained bycrosslinks and entanglements, modeled by a harmonictube potential. The crucial aspect of the model is that thefree ends of the chain can relax their orientation. These“dangling ends” extend as far down the chain as the firstcrosslink, since entanglement constraints on this part ofthe chain may be relaxed by starlike breathing modes [12].For this reason, the length of the dangling end may in factbe several tube diameters.In our model the dangling end fraction is preaveraged,assuming that each chain in the system has two danglingends of identical length, each a fractionf of the totallength of the chain. Since all the averages involving thechain variables are Gaussian, the single chain scatteringfunction is of the formSsqd ­1N2Xl,l0expf2Gqsl, l0dg , (3)whereGqsl, l0d ­12kfq ? srl 2 rl0 dg2l (4)and k· · ·l represents an average over all quenched (i.e.,tube) and annealed (i.e., chain) variables.For the central chain segment confined to the tube, wetake the original Warner-Edwards model (with isotropictube potentials). We find that for any two monomers inthe central tube section,Gqsx, x0d ­ hqsx, x0d ,hqsx, x0d ­XmQ2ml2mjx 2 x0j 1 Q2ms1 2 l2mdd22p6R2g3∑1 2 expµ22p6R2gjx 2 x0jd2∂∏, (5)where we have used the contour length variablesx ­ lyNand x0 ­ l0yN. Note thathqsx, x0d is identical to thexponent in the Warner-Edwards expression (2), althoughthe calculation was done without recourse to the replicatrick, but by diagonalizing the free energy functional (1)using normal modesrp andRp , whererl ­Xprp expsilpd ,Rl ­XpRp expsilpd .(6)The weights for theRp are chosen so as to yield thestandard Gaussian chain result for the scattering in thelimit lm ­ 1.Rm,pRm,2p ­l2mb23p2µ1 13p2d42b4∂. (7)Note that theRp depart from the behavior of a Gaussianwalk at short length scales (highp). The quenched andannealed averages can then be performed explicitly usingGaussian integrals.For the isotropic dangling ends, we take standardGaussian chain statistics. For any two monomers in thesame dangling end,Gqsx, x0d ­ kqsx, x0d ,wherekqsx, x0d ­XmQ2mjx 2 x0j .(8)There is no correlation between the distributions of thedangling ends and the central tube segment. This meansthat for monomer pairs in different chain sections wecan always splitGqsx, x0d into separate contributions fromeach chain section between the two monomers. Writingthe sum over monomers in (3) as an integral, we find thatthe scattering function isSsqd ­ 2Z 10dxZ x0dx0 expf2Gqsx, x0dg , (9)whereGqsx, x0d ­8>>>>>><>>>>>>:kqsx, x0d, for hx, x0j , f.kqs f, x0d 1 hqsx, fd, for f , x # 1 2 f, x0 # f.kqs f, x0d 1 hqs1 2 f, fd 1 kqsx, 1 2 fd, for 1 2 f , x, x0 # f.hqsx, x0d, for f , hx, x0j # 1 2 f.hqs1 2 f, x0d 1 kqsx, 1 2 fd, for 1 2 f , x, f , x0 # 1 2 f.kqsx, x0d, for 1 2 f , hx, x0j.(10)To illustrate that our model does indeed producelozenges, we present in Fig. 1 a fit to the data of Straubeet al. [1], obtained at a stretch ofl ­ 2.9. The lozengeshape is reproduced extremely well. The pattern canqualitatively be considered as a superposition of theisotropic scattering from the undeformed material and theelliptic scattering from the stretched chains, with crosscorrelations. The dangling end fractionf ­ 0.23 required88VOLUME 79, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 7 JULY 1997FIG. 1. Contour plot of a fit to the data of Straubeet al. for lz ­ 2.9, lx ­ ly ­ 0.587, and Rg ­ 142 Å.Fitting parameters weref ­ 0.23 and d ­ 24 Å. Contoursare at intensities of (from the outside) 1.6, 3.3, 6.9, 14, and 30in arbitrary units.to fit that data is a little higher than expected frommeasurements of chain mass between crosslinks. This isto be expected because of the additional effect of chainscission and interactions with unlabeled chains, both ofwhich are present in the Straube system.Futhermore, our model reproduces precisely the obser-vation of an “isotropy angle” [13]. It is found that fora specific angle between the scattering vectorq and thestretch direction, the scattering is to a good approximationthe same as that of the undeformed system. Experimen-tally [1], this angle is found to be given by the conditionXmq2ml2m ­Xmq2m . (11)Substituting this condition into our expression (10) wefind thatGqsx, x0d ­XmQ2mjx 2 x0j for all hx, x0j , (12)which is the value ofGqsx, x0d for an undeformed chain.The reproduction of the isotropy angle is illustrated inFig. 2, which is a contour plot of the difference betweenthe stretched and undeformed scattering functions. The“zero contour” consists of two straight lines at a fixedangle to the stretch direction. We note that this featureis reproduced only under the assumption that the tubepotential does not couple to the strain.The dangling end model above is appropriate for net-works of the type used by Straubeet al. [1], formedby random crosslinking along the melt chains. Danglingends may be partially eliminated by tailoring the reac-tion chemistry to produce specific end linking of chains,though complete elimination is impossible, due to incom-FIG. 2. Contour plot of the differenceS 2 S0 between thestretched and undeformed scattering functions for the samesystem as Fig. 1.plete reaction. We are not presently aware of any rele-vant scattering experiments on pure networks of this lattertype. Such experiments would constitute a strong test ofour present theory.We now briefly discuss the lozenges observed in therecent butterfly scattering experiments [2,3]. The butterflyis the name given to a contour plot where the contourshave a characteristic “figure 8” shape, aligned in thestretch direction. Butterflies appear in scattering fromstretched networks swollen by labeled mobile chains.It is generally accepted [3,14] that the appearance ofthe butterflies is related to static inhomogeneities inthe density of network crosslinks or entanglements, andrequires the forced diffusion of mobile chains towardsregions with fewer crosslinks or entanglements.The butterfly patterns do not appear immediately onstretching the system; there is typically a progressionfrom elliptical contours to lozenges to butterflies. Anexplanation of the three shapes must take into accountthe time window in which each pattern is observed. Ina recent experiment [3] it was observed that the ellipsesappear when the system is first stretched and the entiresystem deforms affinely. Lozenges appear at times ofthe order of the orientational relaxation time for theshort mobile chains. Butterflies appear at times a gooddeal greater than this, when the mobile chains have hadsufficient time to diffuse several radii of gyration.Clearly the important time scale is the orientationalrelaxation time ts of the short chains. At times ofthe order ofts the short chains will have relaxed theirorientation, but will have had time to diffuse one radius ofgyration at most. We propose that the interaction betweenthe relaxed short chains and oriented network chains issufficient to explain the lozenge shape in the scattering.89VOLUME 79, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 7 JULY 1997The uniform density field approach of Edwards andMcLeish [11] provides some mathematical justificationfor this conjecture. There may also be some additionalcontribution due to dangling ends in the network.Finally, we mention an alternative explanation for thelozenge pattern. It has been proposed [3,15] that thelozenges are due to a mechanism similar to the one whichgives rise to butterfly scattering patterns. It is possiblethat the inhomogeneities in crosslink density give rise tononaffine local deformations which result in anisotropicscattering, even in a pure network. The explanation isthat the lozenge is somehow a superposition of ellipticand butterfly scattering patterns. We believe that thisidea arose historically because lozenges were often seenin experiments as a precursor to butterflies. It shouldbe noted, however, that in these butterfly patterns thezero wave vector limit of the scattering is multivaluedand depends upon the direction of the scattering vector[15,16]. This does not appear to be a characteristic featureof the lozenge pattern. Furthermore, this model willnot in general reproduce the isotropy angle observed indeformed pure networks.In conclusion, we find that models involving a propercombination of isotropic and deformed chains will yieldlozenge scattering patterns. This provides a reasonableand consistent explanation for the neutron scatteringpatterns observed in deformed polymer systems.D. J. R. gratefully acknowledges the financial supportof ICI plc. We thank Straubet al. for kindly providingus with their data and W. Pyckhout-Hintzen for usefuldiscussions.[1] E. Straubeet al., Phys. Rev. Lett.74, 4464 (1995).[2] F. Bouéet al., Colloid Polym. Sci.269, 195 (1991).[3] C. Hayeset al., Macromolecules29, 5036 (1996).[4] See, e.g., R. T. Deam and S. F. Edwards, Philos. Trans.R. Soc. London A280, 317 (1976).[5] M. Warner and S. F. Edwards, J. Phys. A11, 1649 (1978).[6] S. Westermannet al., Macromolecules29, 6165 (1996).[7] E. Straube, V. Urban, W. Pyckhout-Hintzen, andD. Richter, Macromolecules27, 7681 (1994).[8] G. Heinrich and E. Straube, Polym. Bull.17, 255 (1987).[9] D. J. Read and T. C. B. McLeish (to be published).[10] J. des Cloizeaux, J. Phys. I (France)4, 539 (1993).[11] S. F. Edwards and T. C. B. McLeish, J. Chem. Phys.92,6855 (1990).[12] R. C. Ball and T. C. B. McLeish, Macromolecules22, 1911(1989).[13] F. Boué, Adv. Polym. Sci.82, 47 (1987).[14] C. Roufet al., Phys. Rev. Lett.73, 830 (1994).[15] S. V. Panyukov, JETP Lett.75, 347 (1992).[16] A. Ramzi, F. Zielinski, J. Bastide, and F. Boué, Macro-molecules28, 3570 (1995).90

‘‘Lozenge’’ contour plots in scattering from polymer networks

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``Lozenge'' Contour Plots in Scattering from Polymer Networks

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‘‘Lozenge’’ contour plots in scattering from polymer networks

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