An investigation into the relationship between ‘gel-effective’ and total numbers of crosslinks in irradiated LLDPE

Polymer International 44 (1997) 300È310

An Investigation into the Relationshipbetween ‘Gel-effective’ and Total

Numbers of Crosslinks in IrradiatedLLDPE¤

Richard A. Jones,* David J. Groves & Ian M. Ward

IRC in Polymer Science and Technology, Department of Physics, University of Leeds, Leeds LS2 9JT, UK

(Received 24 February 1997 ; accepted 17 June 1997)

Abstract : Rheological data for crosslinked LLDPE samples have been related toprevious studies of similar materials. It has been found possible to attribute arange of time constants and characteristic moduli to the “gel-networkÏ(j0in ) (G0i)strength factor (S) of all these materials. Comparability established, the numberof crosslinks estimated rheologically are compared with those computed bymolecular modelling for equivalent gel fractions. Below about 0É4 gel fraction, allcrosslinks are found to be e†ective in forming the gel. Beyond about 0É85 gelfraction, further increase in crosslinks produces only small changes in moduli ;this is observed at around Ðve crosslinks per pre-irradiated molecule. A simplepower relationship is found between the rheological and “gel-e†ectiveÏ numbers ofcrosslinks. A previously postulated relationship between the gel fraction andnumbers of “gel-e†ectiveÏ crosslinks appears to be universal for LLDPEs. “Gel-networksÏ arise at low gel fractions. Rheological data suggest that there is adose-related progression of “gel-networkÏ, from initial viscoelastic polymer,through a medium containing a mobile distribution of “gel-networksÏ of increas-ing size, to a temporary “gel-networkÏ, still able to relax, and Ðnally a saturatedpermanent network.

Polym. Int. 44, 300È310 (1997)No. of Figures : 10. No. of Tables : 3. No. of References : 14

Key words : polyethylene (LLDPE), acetylene, electron beam irradiation, meltrelaxation moduli, total crosslinks, atomistic computation, “gel-e†ectiveÏ cross-links

1 INTRODUCTION

Recent atomistic computer-modelling studies on theformation1 and subsequent reaction of amorphous alkylradical pairs in a linear low-density polyethylene(LLDPE)2 have shown that there are considerably morecrosslinks at any given dose than can be accounted forby termination reactions alone. The level of crosslinkingin excess of radicalÈradical termination reactions wasthus explainable in terms of “gel-e†ectiveÏ chain reac-tions. A “gel-e†ectiveÏ chain step at any given dose

is a chain step which creates a crosslink that(NCS, D)

* To whom all correspondence should be addressed.¤ Dedicated to Professor Bob Stepto on the occasion of his60th birthday.

increases the average molecular weight of the macro-molecules, resulting in a corresponding increase in gelfraction. Not all crosslinks in a network system contrib-ute e†ectively to molecular weight increase ; some areintramolecular and others are intermolecularly repeti-tive in this respect. “Gel-e†ectiveÏ chain step curves werefound by dividing the number of radicals required toproduce a given gel fraction by the number actuallygenerated at the same dose. For irradiations in both anatmosphere of acetylene and in vacuo, was foundNCS, Dto follow near-second-order decay with respect to dose(D), with data Ðtting the expression

NCS, D\ K1(D[ Dg)aK1K2(D[ Dg)2a ] 1

(1)

3001997 SCI. Polymer International 0959-8103/97/$17.50 Printed in Great Britain(

Numbers of crosslinks in irradiated L L DPE 301

where is the dose to incipient gelation andDg K1, K2and a are constants. Knowing the relationship between

and dose, it became possible to obtain theNCS, Dnumber of “gel-e†ectiveÏ crosslinks per initial(NXeff)(unirradiated) number-average macromolecule, i.e.

from the equation(NXeff/NM)gel ,ANXeff

NM

Bgel

\ DG(R)

2NCS, DM1 n, 0 (2)

where is the number of pre-irradiated molecules,NMG(R) is the radiation efficiency for free radical pro-duction (2É40 ] 10~7 mol-spins J~1),3 is the pre-M1 n, 0irradiated number average molecular weight (kg mol~1)and DG(R)/2 is the number of radicals generated priorto the chain reactions. Following this, it was possible toobtain gel fraction (g) versus curves. The(NXeff/NM)gelcurves calculated from data obtained following irradia-tions in the presence of acetylene and in vacuo werenearly identical and Ðtted the expression

g \ 2ANXeff

NM

Bgel

GK3 exp

C[bANXeff

NM

Bgel

D] C

H(3)

where b and C are constants.K3 ,This work raises the question as to what extent the

computed number of “gel-e†ectiveÏ crosslinks equates tothe total number of crosslinks in a network at a givendose or gel fraction. The answer to this questionrequires an analytical method such as the determinationof the modulus of the gel, which yields informationabout the numbers of crosslinks generated in radiation-induced chain reactions, as opposed to the degree oftendency towards inÐnite molecular weight or gel frac-tion. Currently, it is not possible to estimate chemicallythe total number of radiation-induced alkyl crosslinksin solid PEs. Hence the information we require can onlybe obtained from the mechanical properties of the irra-diated LLDPE in the rubbery state (i.e. above thepolymer melting point). In principle, this can be done inone of two ways, either by EdwardsÈVilgis analysis ofthe rubberÈelasticity behaviour4h6 or by melt rheologi-cal dynamic viscoelasticity measurements. The formermethod requires stress-loading the LLDPE samples inthe melt. Unfortunately, this was not possible forsamples of low and high gel fractions. The low-gel-frac-tion samples showed creep to failure and therefore didnot reveal a plateau modulus, while higher gel fractionsbroke before an experimentally signiÐcant number ofstress steps could be recorded. Results were obtainedfrom gel fractions of around 0É5, but this is insufficientfor a dose response or gel fraction analysis. This beingthe case, we decided to use the melt rheology technique,where data could be obtained over almost the entire gelfraction range. The LLDPE used in our original com-puter modelling study2 was, however, too thin for rheol-ogy experiments (D0É126 mm). Di†usion of acetyleneand experimental limitations deemed that the samplesshould be 0É3 mm or less. Hence fresh samples were pre-

pared for rheological investigation. Since almost identi-cal gel fraction versus curves were(NXeff/NM)gelobtained for both irradiations in vacuo and in acetylenein the previous experiments, and scissions are not mini-mized during in vacuo irradiation and annealing pro-cesses, the new data in this study were obtained fromirradiation in acetylene only.

In dynamic viscoelasticity measurements it is usual tocalculate plateau moduli in shear either from high-(GN0)frequency moduli G(u) (G@ and GA values) or from relax-ation moduli (G(t)). can be related to the meanGN0molecular weight between combined entanglements andcrosslinks The combined number of entangle-(MEC).ments and crosslinks is given by the standard(NEC)relationship7

GN0 \NEC kT \ oRTMEC

(4)

where k and R are the Boltzmann and universal gasconstants respectively, T is the absolute temperatureand o is the density of the polymer. There is the possibledifficulty that the PE may crystallize at the measure-ment temperature required to determine Every pre-GN0 .caution was taken to avoid this problem. In addition,

obtained by this method includes componentsGN0resulting from both the crosslinks and the residualentanglements. Overall, the values are probablyGN0greater than the true values representing the e†ects ofcrosslinks alone. Only in the limit of relaxation, at timeslong enough for the entanglements to have relaxed, isthe matrix free from the e†ects of non-topologicallytrapped entanglements. At this time the crosslinksbecome the limiting criterion and we can make theassumption that represents the componentG(t)limmodulus attributable to the crosslinks. This being thecase, it is possible to use the “entanglement-freeÏ moduliof irradiated samples to estimate the number of cross-linked chains which approximates to the true(Nc),number of crosslinks per cubic metre, in the(NX)respective crosslinked networks :

NX B Nc BG(t)limkT

(5)

However, at this time the polymer matrix may still be,e†ectively, a blend of crosslinked and uncrosslinkedregions, because co-operative relaxation may extend thetime scales of the uncrosslinked chains. Winter andChambon8,9 have expressed the relaxation modulus forcrosslinked polydimethylsiloxanes (PDMS) in terms of“gel strengthÏ (S) (which we shall call “gel-networkÏstrength) and time (t), with a relaxation exponent (n),such that

G(t) \ St~n (6)

However, note that although their single materialparameter S (with units Pa sn) provides a sufficientdescription of “gel-networkÏ relaxation for some PDMS

POLYMER INTERNATIONAL VOL. 44, NO. 3, 1997

302 R. A. Jones, D. J. Groves, I. M. W ard

polymers, a viscoelastic material more usually requiresa relaxation spectrum H(j). “Gel-networkÏ here is takento mean networked entanglement in the melt asopposed to gel fraction. Viscoelasticity at the gel pointfor PDMS8,10 and LLDPE11,12 has been Ðtted to Eqn(6), and S has been further expressed by Valles et al. (forLLDPE),11 Scanlan and Winter (for PDMS)10 andHalley and Mackay (for LLDPE)12 in the form

S \ G0 j0n (7)

where the relationship between S and n is a singlemaster curve, independent of crosslink density, is aj0time constant or characteristic time and is a charac-G0teristic modulus which approximates to the plateaumodulus in the case of PDMS.10 Equations (6) and (7)combine as

G(t) \ G0A tj0

B~n(8)

However, data for LLDPE12 indicate that there may bemore than one time constant. We use these equations tocompare experimental data from this and other LLDPEstudies and then relate the total numbers of crosslinksdetermined rheologically with those that e†ectively con-tribute to the increasing macromolecular weight asdetermined by an atomistic analysis of gel fraction data.

2 EXPERIMENTAL

2.1 Materials

LLDPE sheets (BP Inovex LL0209AA9381X) were hotpressed (c. 20 tons for 2 min at 180¡C), quench cooled incold water and cut into discoid samples (thickness c.0É25 mm, diameter c. 25 mm, density c. 0É939 g cm~3(gradient column), kg mol~1,M1 n, 0 B 30É50 M1 w, 0 B 138kg mol~1, polydispersity c. 4É5 (RAPRA TechnologyLtd)).

2.2 Irradiation method

The method used to irradiate the LLDPE samples, inthe presence of acetylene, at the University of Leeds,Cookridge Radiation Research Centre, has beendescribed well previously.13 Simply stated, the sampleswere vacuum degassed (1 h), gassed with acetylene (1 h)prior to electron beam irradiation (room temperature)and then annealed (100¡C, 1 h), still in the presence ofacetylene, to ensure complete reaction of all the rad-icals. Gel fractions of the irradiated samples wereobtained by the method outlined in the same reference.Samples of acetylene-impregnated LLDPE irradiatedwith 1É032, 1É415, 2É249, 3É752, 6É862, 19É476 and32É102 ] 104 Gy (corresponding to gel fractions of0É092, 0É333, 0É542, 0É666, 0É801, 0É923 and 0É961

respectively) were used for the rheological and atomisticanalysis of gel fractions.

2.3 Rheological measurements

All rheology measurements were made by rotationalrheometry using a Rheometrics Dynamic Analyser(RDA2) or a Rheometrics Dynamic Spectrometer(RDS2). In both cases the sample geometry was that ofa cylinder or disc between parallel disc platens (10 mmdiameter) with shear deformation applied about thecylinder axis of symmetry, either in oscillation to deter-mine the in-phase and quadrature dynamic moduli G@and GA, or as a step strain to obtain G(t). Irradiatedsamples were form-stable and would not conform to theclassic cone and plate geometry. Surface-cleansed discsof the LLDPE (10 mm diameter) were laminatedtogether in the rheometer to form cylindrical samples(2É5È3É0 mm thick). All measurements presented herewere made in a nitrogen atmosphere at 160¡C (433 K).Maximum shear strains at the sample circumferencewere between 0É01 and 0É05, in which range all sampleswere linearly viscoelastic in the sense of good sine waveamplitude and strain-independent data. Some sampleswere measured at a range of temperatures down to115¡C in order to obtain a better approximation for

However, in view of the need for entanglements toGN0 .relax, this proved unhelpful, and only the 160¡C dataare used.

3 RESULTS AND DISCUSSION

3.1 Interpretation of rheological data

The angular frequency dependence of the dynamicmoduli G@ and GA at 160¡C provides a viscoelastic Ðn-gerprint of the LLDPE and the e†ect of irradiation.Examples of polymer sheet subjected to radiation dosesof 1É032 and 19É476 ] 104 Gy in acetylene are com-pared with the untreated polymer (Fig. 1). The controlpolymer has a classic polyethylene melt response, withthe quadrature or loss modulus dominant at low fre-quencies indicating Ñuid motion with network controlgiving more elastic behaviour at higher frequencies.Using the terminology of Hess et al.,14 the controlpolymer is in a “pre-gel-networkedÏ state, while the1É032 ] 104 Gy sample is just beyond a “critical-gel-networkÏ in the “post-gel-networkedÏ state and the19.476] 104 Gy sample is well into the “post-gel-net-workedÏ state.

The e†ect of radiation-induced crosslinking is moresimply shown by the decay of G(t) with time. For thecontrol LLDPE polymer, G(t) relaxes steadily on thelogÈlog scale, shown by the data points in Fig. 2, fallingby more than three decades to the limit of the Rheo-metrics RDA2 transducer in less than 100 s. Samples of



Fig. 1. Dynamic moduli of irradiated LLDPE samples kg mol~1 in acetylene) : G@ unirradiated control(M1 n, 0 \ 30É50 (ÈÈ>ÈÈ) ;GA unirradiated control G@ 1É032 ] 104 Gy GA 1É032 ] 104 Gy G@ 19É476 ] 104 Gy (- - -(ÈÈ|ÈÈ) ; (È È = È È) ; (È È K È È) ; + - - -) ;

GA 19.476] 104 Gy (- - -) - - -).

the acetylene-impregnated, irradiated LLDPE show alarge decrease in the rate of relaxation of G(t) (Fig. 2),falling from about one decade in more than 103 s at1É032 ] 104 Gy to almost zero relaxation at32É102 ] 104 Gy.

Values of S obtained from Eqn (6) using the data inFig. 2 are plotted versus radiation dose for the irradi-ated LLDPE examined in the present study (Fig. 3).The previously published LLDPE data11,12 are alsoincluded in Fig. 3. From about 20] 104 Gy upwards

Fig. 2. Relaxation moduli of irradiated LLDPE samples kg mol~1 in acetylene) : unirradiated control(M1 n, 0 \ 30É50 (=) ;1É032 ] 104 Gy 1É415 ] 104 Gy 2É249 ] 104 Gy 3É752 ] 104 Gy 6É862 ] 104 Gy 19É476 ] 104 Gy(K) ; (+) ; ()) ; (>) ; (|) ; (…) ;

32É102 ] 104 Gy lines are theoretical Ðts to Eqns (6)È(8).(L) ;



Fig. 3. S versus irradiation dose for LLDPEs : this study Halley and Mackay12 (grey Valles et al.11 (grey curve Ðt (“all(=) ; |) ; )) ;dataÏ except Ref. 11) is an exponential rise (ÈÈÈ).

the modulus, represented through S, tends to reach alimiting plateau or saturate. This appears to be con-Ðrmed by the data of Halley and Mackay using an irra-diated Dowlex LLDPE.12 The data appear to Ðt theexponential rise of S with dose (S \ 6É6 ] 105[1 [ exp([1É4 ] 10~5] D)]). However, the data of

Valles et al.11 for a laboratory-made, linear, almostmonodisperse, ethyleneÈbutene copolymer, referred toas LLDPE, fall well below the curve for the two com-mercial polymers. Following Valles et al., we haveplotted logS versus n (Fig. 4). We conÐrm the SÈnrelationship as a master curve with the three sources of

Fig. 4. Log S versus n for LLDPEs, reference Eqn (6) : this study Halley and Mackay12 Valles et al.11 (grey high(+) ; (K) ; |) ;slope low slope()) ; (>).



LLDPE data in Fig. 4. Using Eqn (7), Halley andMackay further divided the SÈn data into high and lowlinear regression slopes, depicting two rheological stateswith the two lines intersecting at n B 0É5. This is con-Ðrmed using “all dataÏ in Fig. 4. They referred to then B 0É5 intersection as a “pseudo-gel pointÏ, with a truegel, determined from insoluble fraction, occurring at aradiation dose of c. 3É2 ] 104 Gy (S [ 105). We notethat in Fig. 3 a dose of 3É2 ] 104 Gy places S at thetransition between rising S and the limiting S plateau(point A), whereas the “pseudo-gel pointÏ (n B 0É5) givesS B 1É3 ] 104 on the initial steep rise (point B) andprobably has no signiÐcance. The predicted high-slopevalues for and the characteristic time are given inG0 j0Table 1. Although some di†erences between the threepolymers may be expected, inspection of the data in Fig.3 and 4 indicates that and may be inÑuenced byj0 G0the di†erent levels of radiation-induced crosslinking andhence the slightly di†erent ranges of n and S over whichthe values are determined. However, when comparedwith the “all dataÏ low-slope characteristic time of2É5 ] 10~2 s, below the “pseudo-gel pointÏ, the tabulatedhigh-slope data are clearly a group.

Scanlan and Winter10 with PDMS and Valles et al.11with monodisperse “LLDPEÏ have obtained good linearÐts of the “gel-networkÏ strength (log S) and relaxationexponent n using Eqn (7). However, with commercialpolydisperse LLDPE the present work and the resultsof Halley and Mackay demonstrate a non-linear Ðt oflog S and n in Eqn (7), with the need to take high- andlow-slope time constants as shown in Fig. 4. With theadditional data the Ðt is more clearly a curve, indicatinga range of time constants appropriate to these polydis-perse materials. We obtain the approximate polynomialcurve Ðt to all three sources of LLDPE data (this workand Refs 11 and 12) shown in Fig. 5 using three timeconstants in the form

S \ G01 j01n ] G02 j02n@2]G03 j03n@3 (9)

where and represent decreasing moduli atG0i j0iincreasing characteristic times. The dominant Ðrst termgives values for the time constant and modulus of

and respectively,j01\ 6É4 ] 10~5 G01 \ 7É1 ] 105,while the second-term values are andj02 \ 7É9 ] 10~3

The continuous decrease in S withG02\ 7É9] 104.increasing n and the possible continuous distribution of

and suggest that an increasing radiation doseG0i j0i

TABLE 1. High S–n slope-predicted andG0

k0

values for Eqn (7)

Slope region (source) l0

(s) G0

(Pa)

High (this work) 2·7 Ã10É5 8·1 Ã105

High (Ref. 11) 1·4 Ã10É4 9·3 Ã105

High (Ref. 10) 6·4 Ã10É4 7·4 Ã105

High (all) 2·5 Ã10É4 7·1 Ã105

may give a progression of “gel-networkÏ formation. It isworthy of note that the Ðrst-term parameters andG01

are very close to the high-slope linear Ðt values inj01Table 1. We speculate that the initial viscoelasticpolymer melt may progress via a viscoelastic mediumcontaining a mobile distribution of “gel-networksÏ ofincreasing size, to a temporary “gel-networkÏ, which maystill partially relax, Ðnally through to a saturated per-manent network.

The comparison between the measured G(t) for ourseries of irradiated LLDPE (points) and the predictionfrom Eqns (6) and (9) (lines) is shown in Fig. 2, whichdeÐnes a level of conÐdence for the subsequent use ofG(t) in the determination of via Eqn (5). A relativeNXerror for G(t) is calculated as 11% (1É2% in logG(t)).

3.2 Rheological determination of networkcrosslinking

It can be seen from Fig. 2 that the entanglementcontribution to G(t) of the non-crosslinked controlhad relaxed to the limit of reliable measurement by about33 s. By invoking the basic assumptions described inEqn (5) and using of the irradiated samples,G(t) ] G(t)limthe equivalent numbers of crosslinks per initial, pre-irradiated, number-average molecule, i.e. in(NX/NM)rh ,the respective crosslinked networks were calculated.(The number of initial pre-irradiated molecules percubic metre is m~3,NM\ oNa/M1 n, 0\ 1É854 ] 1025where o is the density (0É939 g cm~3), is the pre-M1 n, 0irradiated number-average molecular weight and isNaAvagadroÏs number.) Unfortunately, it is not possible todetermine directly the time at which the entanglementcontribution to G(t) of the control had completelyrelaxed. However, it can be seen that the moduli values,prior to the limit of measurement, decreased with timeas a linear logÈlog function. Hence values(NX/NM)rhwere calculated for each of the irradiated samples ofacetylene-impregnated LLDPE, at half-decade intervalsfrom 3É16 ] 10 to 1 ] 103 s, in order to determine whenthe e†ects of entanglements may be neglected. Theresults are compared with equivalent (NXeff/NM)gelvalues in Section 3.4.

3.3 Computational determination of ‘gel-effective’crosslinking

The gelÈdose curve for the LLDPE kg(M1 n, 0B 30É50mol~1) used in these experiments is shown in Fig. 6,together with the results for the LLDPE (M1 n, 0 B 22É05kg mol~1) obtained from our previous computationalexperiments.2 Clearly there is only a very small, but sig-niÐcant, di†erence between the two curves. Hence, aftera lower macromolecule size limit, further changes in

are not expected to alter the gel fraction yield sig-M1 n, 0niÐcantly. Also included in Fig. 6 are the computedDG(R)/2 curves for the respective LLDPEs, which rep-



Fig. 5. Log S versus n for LLDPEs reference Eqn (9) : this study (grey Halley and Mackay12 Valles et al.11 (grey)) ; (K) ; |) ;third-degree polynomial Ðt (ÈÈÈ).

resent the maximum obtainable gel fractions at anygiven dose without the intervention of chain reactions,i.e. if only radicalÈradical termination reactionsoccurred with no main-chain scissions.

The numbers of “gel-e†ectiveÏ chain steps for(NCS, D)the two LLDPEs, as determined by the methoddescribed in Section 1 and more fully in the previouspublication,2 are shown as a function of dose in Fig. 7.It can be seen quite clearly that for theNCS, D M1 n, 0 B

kg mol~1 LLDPE is lower at any given dose than30É50for the kg mol~1 LLDPE. Since there isM1 n, 0B 22.05

Fig. 6. Gel fraction versus dose plots : kg mol~1M1 n, 0 \ 30É50in acetylene (this study) computed maximum gel(È >ÈÈ) ;fraction curve (DG(R)/2) for kg mol~1 (thisM1 n, 0\ 30É50study) (- - - - - -) ; kg mol~1 in acetylene2,13M1 n, 0 \ 22.05

computed maximum gel fraction curve (DG(R)/2)(È È L È È) ;for kg mol~1 (É É É É É É).M1 n, 0\ 22É05

little di†erence in gel fraction at any given dose for thetwo polymers (Fig. 6), the curves are probably close tothe maximum obtainable These data demon-NCS, D .strate that a lower number of “gel-e†ectiveÏ chain stepsis required from the LLDPE, chosen forhigher-M1 n, 0the rheological experiments, to form similar gel frac-tions to those of the LLDPE. Apart fromlower-M1 n, 0this, the curves, obtained under identical conditions,decay in an almost identical manner, indicating that thedecays of with respect to dose are close to theNCS, Dexpected second order for radicalÈradical terminations.Estimates for the order of decay with respect toNCS, Ddose, as rth order, where are shown(NCS, D)1~r\ k

rD,

versus dose plots : kg mol~1 inFig. 7. NCS, D M1 n, 0 \ 30É50acetylene (this study) kg mol~1 in(ÈÈ>ÈÈ) ; M1 n, 0\ 22É05

acetylene2 Ðts to Eqn (1) (- - - - -).(ÈÈLÈÈ) ;



to be very similar in Table 2. The least-squares Ðttingparameters for the constants in Eqn (1) are also shownin Table 2, where it can be seen that the constant a isvery similar for the two LLDPEs.

was then calculated via Eqn (2), as(NXeff/NM)geldescribed in Section 1, for the gel fractions determinedfor the LLDPE samples used in the rheologi-high-M1 n, 0cal assay of and the results are compared in(NX/NM)rhSection 3.4.

3.4 Comparison of rheological and gel-effective’crosslinks

In order to examine the relationship between thenumber of crosslinks determined rheologically and the“gel-e†ectiveÏ number estimated computationally, dosefor dose, or gel fraction for gel fraction, was(NX/NM)rhplotted against The result is very inter-(NXeff/NM)gel .esting. Firstly, the isochronal curves (Fig. 8(a)), resultingfrom the half-decade intervals of G(t), converge to thelimiting “entanglement-freeÏ relaxation time, whichpasses through the origin of the graph. Hence it can beseen that the last half-decade time interval (1] 103 s) is,within experimental error, truly representative of thecomplete entanglement-free relaxation time and there-fore s).G(t)limBG(1] 103

The gradient of the initial portion of thisentanglement-free curve (Fig. 8(a)), below about 0É3

is almost exactly unity, as can be seen(NXeff/NM)gel ,from the superimposed line. Thus below 0É3 “gel-e†ectiveÏ crosslinks per pre-irradiated number-averagemacromolecule (c. 0É4 gel fraction), (NX/NM)rh\

and all crosslinks are e†ective in forming(NXeff/NM)gelthe gel. The basic premise of the atomistic approach tothe computational determination of “gel-e†ectiveÏ cross-linking is that only one crosslink is required betweentwo previously unlinked macromolecules for the newlylinked macromolecules to contribute to the gel fraction.Quite clearly, this premise is corroborated by the initialgradient being equal to unity.

As far as the last data set in Fig. 8(a) (0É912or 0É801 gel fraction) is concerned, the(NXeff/NM)gel

“entanglement-freeÏ curve can be Ðtted to the power

relationship

ANXNM

Brh

\ANXeff

NM

Bgel

] q2ANXeff

NM

Bgel

q(10)

where the power constant q approximates to the integerfour, as shown by the solid curve. Although this equa-tion may not be an exact solution to the true rela-tionship between and it Ðts(NX/NM)rh (NXeff/NM)gel ,the data well and is indicative of a statistical expan-sion, where we would expect to Ðnd integer powerfunctions. It can be seen from the extended curve inFig. 8(b) that somewhere between 0É912 and 1É433

(0É801È0É923 gel fraction), Eqn (10) is no(NXeff/NM)gellonger valid. It is not statistically logical that the

gradient should begin to be(NX/NM)rh/(NXeff/NM)gelreduced in this region. The gradient should alwaysincrease as more and more real crosslinks are requiredper “gel-e†ective“ crosslink. The only logical explanationis that in this region an increasing proportion of the realcrosslinks no longer contribute to G(t) and increasingly

The stress no longer passes through allNX [ G(t)lim/kT .the crosslink pathways and crosslink modulus satura-tion results.

Plots of gel fraction (g) versus number of crosslinksper initial number-average molecule are shown(M1 n, 0)in Fig. 9. Firstly, it can be seen from the gel fractionversus plots that the higher LLDPE(NXeff/NM)gel M1 n, 0gives an almost identical curve to that of the lower

LLDPE used in our previous experiments, devi-M1 n, 0ating only slightly from about 0É7 gel fraction onwards.Thus it appears that gives rise to identical(NXeff/NM)gelgel fractions in both LLDPEs, regardless of the initial

or the conditions required to produce the “gel-M1 n, 0e†ectiveÏ crosslinks, because previously the same curvewas obtained for both in vacuo and acetylene-incorporated irradiations of the lower LLDPE.M1 n, 0Slight deviations in the curve for the two LLDPEsprobably arise from experimental errors. This is quitereasonable and we expect the curve to be universal forall oleÐnic polymers and possibly for other similar poly-meric systems. In our previous paper concerning thenature of “gel-e†ectiveÏ crosslinks,2 the lower M1 n, 0LLDPE curves were Ðtted to Eqn (3). The constant b ofEqn (3) approximates to unity and the expression

fitting parameters for Eqn (1) a) andTABLE 2. NCS, D

(K1, K

2,

simple rth-order analysis (kr= (N

CS, D)1—r/D)

M1n, 0

(kg molÉ1) 22·05 (Ref. 2 in C2H

2) 30·50 (this study in C

2H

2)

K1

(GyÉa) 7·93 Ã10É3 3.50 Ã10É3

K2

(GyÉa) 2·30 Ã10É5 3·68 Ã10É5

a 0·811 0·787

Increasing order r 0·68 0·61

Increasing rate kr

2.53 Ã10É4 1.23 Ã10É4

Decreasing order r 2·12 2·52

Decreasing rate kr

1·99 Ã10É6 1·99 Ã10É6



Fig. 8. (a) Rheologically determined versus com-(NX/NM)rhputed plots for half-decade G(t)s of 3É16 ] 10 s(NXeff/NM)gel

1É00 ] 102 s (- - - 3É16 ] 102 s(É É É É L É É É É), K - - -), (È È | È È)and 1É00 ] 103 s the gra-(ÈÈÈÈ) ; (NX/NM)rh\ (NXeff/NM)geldient is shown (È È È). (b) versus computed(NX/NM)rh

plot showing extended Ðt to Eqn (9) from data at(NXeff/NM)gel1É00 ] 103 s moduli-saturated data sets at 1É433 and(ÈÈÈ);1É875 are shown using the same time symbols(NXeff/NM)gelas in (a) ; the gradient is shown(NX/N

M)rh\ (NXeff/NM)gel

(È È È).

reduces to

g \ANXeff

NM

Bgel

2GK3 exp

C[ANXeff

NM

Bgel

D] C

H(3@)

This equation Ðts the computed data for the twoLLDPEs extremely well, yielding the least-squaresÐtting parameters given in Table 3.

It is possible that is ln(2), but the signiÐcance ofK3the constant C is not immediately apparent. The equa-tion is by no means an ultimate deÐnition of the situ-ation, but the signiÐcance of it can be viewed from thefollowing description of the gel fraction :

g \ ;i/1i/=/

i;

j/1j/MNj

(11)

Fig. 9. Gel fraction versus crosslinks per molecule plots :M1 n, 0plot for kg mol~1 in vacuo and in(NXeff/NM)gel M1 n, 0 \ 22É05

acetylene (. . . . . .) ; plot for kg mol~1 in acetyleneM1 n, 0 \ 30É50(È È È È) ; plot for kg mol~1 in acety-(NX/NM)rh M1 n, 0 \ 30É50lene from plot treated with Eqn (10) (ÈÈÈ);(NXeff/NM)gel

data points for kg mol~1 in acetylene(NX/NM)rh M1 n, 0 \ 30É50hypothetical 2 gradient (- - - - -).(…) ; (NXeff/NM)gel

where is the functionality of the ith “gel-e†ectiveÏ/i

crosslink and is the jth molecule ; is always two atMj

/i

the start of a “gel-networkÏ group and unity thereafter.At the summation limit for the particular gel fraction,

is in fact and the expression can be rewritten as;Nj

NM

g \ANXeff

MM

Bgel

;i/1i/=/

iNXeff

\ANXeff

NM

Bgel

' (12)

where ' is the mean functionality of all the “gel-e†ectiveÏ crosslinks in all the combined “gel-networkÏregions. Obviously, ' should be two for inÐnitely smallgel fractions, tending, rapidly at Ðrst and then slowly, tounity at a gel fraction of unity. Thus exp2MK3

in Eqn (3@) is equal to ' and rep-[[(NXeff/NM)gel]] CNresents this tendency. The broken-line gradient

in Fig. 9 represents the hypothetical non-2(NXeff/NM)gelrandom placement of the minimum number of “gel-e†ectiveÏ crosslinks to achieve a gel fraction suchthat each molecule is only connected once, i.e. when

TABLE 3. Gel least-squares fitting parameters for

Eqn (3º)

M1n, 0

(kg molÉ1) K3

C

22·05 (in vacuo) 0·7105 0·1612

(0·6931)a (0·1670)

22·05 (in C2H

2) 0·6928 0·1681

(0·6931)a (0·1680)

30·50 (in C2H

2) 0·7135 0·1494

(0·6931)a (0·1546)

fixed at ln(2).a K3



Fig. 10. Crosslinks per (30É50 kg mol~1) moleculeM1 n, 0versus dose plots (in acetylene) : plot from(NXeff/NM)gel NCS, Dversus dose data (Fig. 7) and Eqn (2) (È È L È È) ; (NX/NM)rhplot obtained from plot treated with Eqn (10)(NXeff/NM)gel

data points equivalent(ÈÈ|ÈÈ); (NX/NM)rh (. . . > . . .) ;data points to rheology study(NXeff/NM)gel (…).

It can be shown2MK3 exp[[(NXeff/NM)gel]] CN\ 2.from the Ðtting parameters to Eqn (3@) in Table 3 thatthis functionality never quite reaches two, even below0É3 where Thus “gel-net-(NXeff/NM)gel NX \ NXeff .worksÏ rather than multiple, disconnected, single cross-links are formed early on in the curve. This is asmight be expected from radiation-induced crosslinkingsystems, where the crosslinks are created via chainreactions which occur even in the absence of unsaturatedbridging agents. Both LLDPEs appear to betending to at gel fraction unity.(NXeff/NM)max\ 2This may be explained by a network where the average

chain is connected to four adjacent neighboursM1 n, 0(The actual value of(NXeff/NM\ 4/(1 ] 4/4) \ 2).

need not be an integer, since it is an(NXeff/NM)maxaverage.)

The predicted gel fraction versus curve(NX/NM)rhobtained by treating the computed with(NXeff/NM)gelEqn (10) is also shown in Fig. 9 (solid curve), togetherwith the actual gel fraction versus data(NX/NM)rhpoints. The two highest data points greater than 0É85gel fraction do not fall on the curve, because of the satu-ration of modulus values at high crosslink density asdescribed earlier. The curve Ðts the remaining datapoints well, which is reasonable proof of the form ofEqn (10), because for gel fractions below 0É85, slightdeviations from the data set are magniÐed in going fromFig. 8(b) to Fig. 9. Earlier attempts to transfer exponen-tial and linear Ðts from the versus(NX/NM)rh

data to the gel fraction versus(NXeff/NM)gel (NX/NM)rhcurve resulted in large errors. The gel fraction versus

curve of Fig. 9 is not likely to be universal,(NX/NM)rhunlike the gel fraction versus curve,(NXeff/NM)gelbecause deviations in physical structure may well a†ectthe number of real crosslinks required to achieve a

given gel fraction. Hence, although the form of Eqn (10)is thought to be approximately general, the powerfunction q of the second term may di†er from one typeof polymer to another. It may even be necessary to addsubsequent power terms in order to satisfy di†erentaverage functionalities of di†erent poly-(NXeff/NM)maxmers.

Finally, in order to have a better feeling for the quan-tiÐcation of crosslinking, and(NX/NM)rh (NXeff/NM)gelare plotted as a function of dose from data obtained inthis study (in acetylene ; Fig. 10). The diagram clearlyshows the massive increase in redundancy of true cross-links with increasing dose as “gel-e†ectiveÏ crosslinks areformed only very slowly. Again the two highest

data points deviate from the extended predic-(NX/NM)rhtion curve because of saturation of the modulus at highcrosslink densities, which appears to limit moduli valuesat about Ðve crosslinks per initial molecule.(NX) M1 n, 0

4 CONCLUSIONS

Rheology data suggest that increasing irradiation dosemay produce a continuous and progressive develop-ment of “gel-networkÏ, from the initial mobile visco-elastic polymer, through a medium containing a mobiledistribution of “gel-networksÏ of increasing size, to atemporary “gel-networkÏ, still able to relax, and Ðnallyto a saturated permanent network. The relaxationmodulus values G(t) for our series of radiation-crosslinked LLDPEs are in close agreement with thepublished data of other authors, following previouslysuggested Ðtting procedures. It has been found possibleto attribute a range of time constants and charac-(j0in )teristic moduli to the “gel-networkÏ strength factor(G0i)(S) for the polydisperse commercial materials used inthis and the reviewed studies. The results of our rheolo-gical estimation of gel fractions should therefore bebroadly representative for LLDPEs.

The “entanglement-freeÏ s). BelowG(t)limB G(1] 103about 0É4 gel fraction, and all(NX/NM)rh\ (NXeff/NM)gelcrosslinks are e†ective in forming the gel. This corrobo-rates the atomistic computational model in that onlyone crosslink is required between two previouslyunlinked macromolecules for the newly linked macro-molecules to contribute to the gel fraction. A simplepower relationship, indicative of a statistical expansion,is proposed to exist between and(NX/NM)rh

This relationship is not likely to be uni-(NXeff/NM)gel .versal, because deviations in physical structure may wella†ect the number of real crosslinks required to achievea given gel fraction ; hence the power function maydi†er from one type of polymer to another. Beyondabout 0É85 gel fraction an increasing proportion of thereal crosslinks no longer contribute to G(t) and cross-link modulus saturation results at around Ðve crosslinksper initial molecule.M1 n, 0

Almost identical curves were found forgelÈNXeff/NM



two di†erent LLDPEs and 30É50 kg(M1 n, 0\ 22É05mol~1) from irradiations both in vacuo and in acetylene,indicating a universal curve for LLDPE, and possiblyother similar polymeric systems, conforming to the pre-viously postulated relationship2 (Eqn (3@))gelÈNXeff/NMin terms of mean functionality and numbers of “gel-e†ectiveÏ crosslinks. As expected for chain reactionsresulting from radiation-induced crosslinking, “gel-networksÏ rather than multiple, disconnected, singlecrosslinks arise at extremely low gel fractions. Thereappears to be an average of about four “gel-e†ectivelyÏconnected pre-irradiated macromolecule equivalents inthe network per initial macromolecule at gel fractionunity.

ACKNOWLEDGEMENTS

R.A.J. was funded by EPSRC (Grant No GR/L62306)and the Hoechst-Celanese Corporation.

DEDICATION

This work is dedicated to Professor R. F. T. Stepto onthe occasion of his 60th birthday anniversary (may hehave many more of them!). We would like to thank Bob

particularly for the many hours of discussion and col-laboration he has given us. We must also apologize tohim for being consistently vague when quizzed aboutthe work covered here. We hope this makes up for it.Happy birthday Bob.

REFERENCES

1 Jones, R. A., Taylor, D. J. R., Stepto, R. F. T. & Ward, I. M., J.Polym. Sci., B; Polym. Phys., 34 (1996) 901.

2 Jones, R. A., Taylor, D. J. R., Stepto, R. F. T. & Ward, I. M.,Polymer, 37 (1996) 3643.

3 Ichikawa, T., Kawahara, S. & Yoshida, H., Radiat. Phys. Chem., 26(1985) 731.

4 Ball, R. C., Doi, M. Edwards, S. F. & Warner, M., Polymer, 22(1981) 1010.

5 Edwards, S. F. & Vilgis, Th., Polymer, 27 (1986) 483.6 Brereton, M. G. & Klein, P. G., Polymer, 29 (1988) 970.7 Ferry, J. D., V iscoelastic Properties of Polymers, 2nd edn. Wiley,

New York, 1970, p. 404.8 Chambon, F. & Winter, H. H., J. Rheol., 31 (1987) 683.9 Winter, H. H. & Chambon, F., J. Rheol., 30 (1986) 367.

10 Scanlan, J. C. & Winter, H. H., Macromolecules, 24 (1991) 47.11 Valles, E. M., Carella, J. M., Winter, H. H. & Baumgaertel, M.,

Rheol. Acta, 29 (1990) 535.12 Halley, P. J. & Mackay, M. E., Polymer, 35 (1994) 2186.13 Jones, R. A., Salmon, G. A. & Ward, I. M., J. Polym. Sci., B;

Polym. Phys., 31 (1993) 807.14 Hess, W., Vilgis, T. A. & Winter, H. H., Macromolecules, 21 (1988)

2536.


Documents

An investigation into the relationship between ‘gel-effective’ and total numbers of crosslinks in irradiated LLDPE