molar mass distributions of end-linked polystyrene star macromolecules

Polymer International 44 (1997) 380È390

Molar Mass Distributions of End-linkedPolystyrene Star Macromolecules¹

Max Weissmu� llerº & Walther Burchard*

Institute of Macromolecular Chemistry, University of Freiburg, D-79104 Freiburg, Germany

(Received 11th March 1997 ; accepted 17 June 1997

Abstract : Polystyrene three-arm star molecules with OH tips were prepared byliving anionic polymerization using an acetal-protected lithium initiator followedby coupling with 1,3,5-triallyl-oxy-2,4,6-triazine and ensuing cleavage of theacetol groups. The functionalized star molecules were end-crosslinked bytoluene-2,4-diisocyanate. Randomly branched samples with various extents ofcrosslinking were prepared under conditions of full conversion of the isocyanategroups. The samples were fractionated analytically by SEC in on-line com-bination with LALLS and a viscosity detector. The mass fraction molar massdistributions could be well Ðtted by the FloryÈStockmayer distribution if thedegree of crosslinking was X ¹ 6. A broad region of crossover to percolationbehaviour followed and the system reached its asymptotic region for X [ 200.Power-law behaviour was already obtained at X B 70, but the exponent qexp\2É36 was found to lie between for the mean Ðeld approximation andqFS \ 2É5

for the percolation prediction.qperc\ 2É2

Polym. Int. 44, 380È390 (1997)No. of Figures : 10. No. of Tables : 2. No. of References : 31

Key words : end-linked star molecules, mean Ðeld approximation, percolationtheory, FloryÈStockmayer distribution, crossover from mean Ðeld to percolationregime, SEC/LALLS/VISC

INTRODUCTION

A striking feature of most macromolecules is theoccurrence of a size distribution. Under common chemi-cal conditions an ensemble of strictly uniform species isnever obtained. This fact was Ðrst observed with linearchains when molar mass determinations by end-groupanalysis or osmotic pressure measurement gave valuesabout two times smaller than by ultra centrifugation orlight scattering. The two deviating molar masses weresoon recognized as number-average and weight-M1 n

* To whom all correspondence should be addressed.¤ Dedicated to Professor Bob Stepto on the occasion of his60th birthday.” Current address : Hoechst AG, CRT Analytical Laboratory,D-65926 Frankfurt, Germany.

average of a well-deÐned molar mass distribution.M1 wBecause of its universal character, Flory denoted thisdistribution as a “most probableÏ1 one. Later scientistspreferred to speak of the SchulzÈFlory distribution inappreciation of their independent but fundamentalwork on the kinetics of polymerization.1,2

The determination of the two averages andM1 w M1 nand characterization of a polymer system by the hetero-geneity ratio or the molecular non-uniformityM1 w/M1 n

were soon considered insufficient. Theu 4M1 w/M1 n [ 1non-uniformity is related only to the width of the dis-tribution by the relationship where p2u \ p2/M1 n2 ,denotes the mean square deviation of the mass from themean Thus u gives no information on the shape ofM1 n .the distribution. However, the shape of a most probabledistribution obtained, for instance, by free radical poly-merization di†ers signiÐcantly from that of a Poisson

3801997 SCI. Polymer International 0959-8103/97/$17.50 Printed in Great Britain(

End-linked polystyrene star macromolecules 381

distribution obtained by living anionic poly-merization.3h5

Various types of distributions could be deriveddirectly from the kinetic scheme, and the experimentaldetermination of molar mass distributions was soonregarded as a valuable means of elucidating the mecha-nism of polymerization.1,6,7 The kinetic equations inpolymerization reactions are highly coupled, and some-times sophisticated mathematical techniques had to beapplied to solve the problem. The Laplace transform-ation and numerical solution on a computer are prob-ably the most widely used techniques.6

The problem became signiÐcantly simpliÐed whenFlory1 introduced the extent of reaction, a, which isdeÐned as the ratio of reacted functional groups in apolymer ensemble to the sum of reacted and non-reacted groups :

a \ reacted groupsreacted] non-reacted groups

(1)

This extent of reaction can never exceed the value ofunity and is thus a probability of reaction. Thereforecommon rules of probability theory can be applied.8For instance, for linear chains obtained by poly-condensation or free radical polymerization, Floryimmediately derived his famous most probabledistribution1a,2

w(n) \ (1 [ a)2nan~1 (2)

with

M0/(1 [ a)\ M1 n B M1 w/2 (3)

where n is the number of bonds (degree ofpolymerization) in the chain.

The corresponding size distribution of branched poly-mers obtained by random polycondensation of f-functional monomers is more involved1b and was Ðrstderived by Stockmayer9 with the result

w(n) \ (1[ a)2a

f[( f [ 1)n] !

(n [ 1) ![( f [ 2)n ] 2] !bn (4)

where

b \ a(1[ a)f~2 (5)

Again a is related to the number or weight-averagedegree of polymerization :

M1 nM0

\ (1 [ a f/2)~1

M1 wM0

\ 1 ] a1 [ ( f [ 1)a

(6)

Equations (4)È(6) reduce to those for the most probabledistribution when f\ 2.

For large degrees of polymerization the Stirlingapproximation n ! \ (2n)1@2(n/e)n can be applied, and

close to the gel point

ac \ 1/( f[ 1) (7)

a further approximation can be made that ledStockmayer9 to the asymptotic distribution

w(n) ](1 [ a)2

af

(b/bc)n2)n( f [ 1)5@2n3@2 (8)

where

bc\( f [ 2) f~2( f [ 1) f~1 (9)

The latter results from Eqn (7) inserted in Eqn (5).

MEAN FIELD APPROXIMATION AND

PERCOLATION

FloryÏs concept of using extents of reaction instead ofkinetic constants proved to be very fruitful and culmi-nated in GoodÏs cascade approach10 that was adaptedto chemistry by Gordon.11 Further simpliÐcations fol-lowed, in particular that by Miller and Macosko,12,13and the extension to conformational properties.14,15The FloryÈStockmayer (FS) approach appeared verynatural, and therefore it remained widely unappreciatedthat a serious approximation was implemented with theassumption of a size- and space-independent probabilityof reaction. Flory16 demonstrated with a series ofexperiments that the reactivity of a functional group isnot a†ected by the size of the macromolecule to whichit is attached. Stockmayer and co-workers,17,18 on theother hand, showed that on diluting the reactionmedium, intramolecular reactions are enhanced, butthis was believed to shift only the critical point ofgelation to higher extents of reaction than predicted byEqn (7).

It took more than 25 years before physicists19,20 re-cognized that FloryÏs approach corresponds formally toa mean Ðeld approximation, and it was argued19h22that the predicted behaviour cannot be correct when thegel point is approached. At this critical point at leastone branched cluster has reached a size that spans thereaction vessel. Now packing problems arise from thefact that the repeating units have a Ðnite volume. Thefractal dimension in the relationship23df

M P Rgdf (10)

should never exceed a value of which would cor-df \ 3,respond to close sphere packing. However, Zimm andStockmayer24 had already demonstrated that FloryÏsconcept eventually leads to when the gel point isdf \ 4approached.

POLYMER INTERNATIONAL VOL. 44, NO. 3, 1997

382 M. W eissmu� ller, W . Burchard

Since self-avoiding reactions could not be treatedanalytically, physicists started to simulate on computersthe reaction by bond percolation on a lattice.21 Amongothers, they established the following asymptoticrelationship for the weight fraction distribution :

w(n) \ An~(qperc~1)f (n/n*), qperc\ 2É2 (11)

and found fractal dimensions for the individual clustersof and 2É5 for swollen and non-swollen clustersdf \ 2É0respectively. On averaging Eqn (10) over the size dis-tribution of Eqn (11), one obtains the ensemble fractaldimension which reduces the valuesdf, e\ df(3 [ qperc),to 1É6 and 2É0 respectively. The di†erence in these twodimensions results from the fact that the molar mass is aweight average but the mean square radius of gyr-(M1 w)ation is a z-average The cut-o† function f (n/n*)(SRg2Tz

).(not further speciÐed) causes a much stronger decay tozero than that due to the power law w(n) P n~(qperc~1)when the degree of polymerization exceeds the value n*.Careful experiments,25 performed later on, gave evi-dence for n* being proportional to the z-average degreeof polymerization, i.e. n* P n

z.

It is of interest to compare the two asymptotic molarmass distributions of Eqns (8) and (11). Both distribu-tions have the same shape, with power-law behaviour inthe intermediate range of n and a cut-o† function atlarge n. Di†erences lie in the value of com-qFS \ 2É5pared with and in the cut-o† function, whichqperc \ 2É2is well deÐned in the FS distribution

fFS(n/n*)\ (b/bc)n B exp[[(1 [ b/bc)n] (12a)

such that

n* \ [1[ (b/bc)]~1

\C1 [ a( f [ 1)

A(1 [ a)( f [ 1)f[ 2

Bf~2D~1(12b)

The corresponding cut-o† function in percolationtheory is not explicitly known. At this point we have torecall that the two asymptotic distributions can only bevalid near the gel point, i.e. when The validityb Bbc .range of Eqn (11) remains unspeciÐed in percolationtheory. It is certainly sensible to assume that the volumeof the repeating unit has no e†ect on the distribution aslong as the degree of polymerization is small, i.e. n > n*,but a transition to percolation behaviour is to beexpected when n ? 1.

At the present stage, theory cannot predict at whichdegree of crosslinking X this transition will occur. Forthis reason we undertook an experimental study, choos-ing OH-terminated three-arm polystyrene star mole-cules as large trifunctional monomer units, which werethen crosslinked by diisocyanate. The size of this three-arm star molecule was sufficiently large (with M1 w \60 000) that even for low extents of reaction the molarmass distribution could be accurately measured by size

exclusion chromatography (SEC) in combination withlow-angle laser light scattering (LALLS) and viscometry(VISC). The present contribution is concerned only withthe molar mass distribution. The e†ect of branching onthe intrinsic viscosity of the individual fractions will bereported in a subsequent paper.26

EXPERIMENTAL

The samples27,28

OH-terminated three-arm polystyrene (PS) star macro-molecules were prepared by living anionic poly-merization of styrene followed by coupling of the livingprecursor with 1,3,5-triallyl-oxy-2,4,6-triazine. The func-tionalized tips of the arms were obtained by using anorganolithium initiator that carried an acetal group onits aliphatic end. After coupling the precursor to form athree-arm star, the acetal groups were cleaved by weakacidic treatment in THF, setting the terminal OHgroups free.

These OH groups were coupled by toluene-2,4-di-isocyanate. Randomly branched samples of increasingdegrees of polymerization (with the star molecule asrepeating unit) were prepared by varying the stoichio-metric ratio in all cases keepingr \ [OH]0/[NCO]0 ,

and taking the reaction to com-[NCO]0\ [OH]0pletion. The number of non-reacted free OH groupswas determined by UV/VIS spectroscopy usingp-nitrophenyl isocyanate as chromophore. Details of thestar synthesis and coupling reactions are given in pre-vious papers.27,28

Size Exclusion Chromatography (SEC) set-up

SEC measurements were made in two arrangements.

1. SEC was coupled with UV and LALLS detectors.Styragel columns (104] 105 Waters) wereÓ,used ; UV absorption was measured at a wave-length j \ 254 nm and LALLS was registeredwith a KMX-6 instrument (Milton-Roy).

2. SEC was coupled with UV, LALLS and VISCdetectors. A linear styragel column (Waters) wasused. The VISC detector K200 was an instrumentof Knauer, Berlin.

The chromatography was driven by an HPLC pump 64(Knauer) in a pressure range of 6È15 bar. The injectionvolume was in both cases 20 ll. Tetrahydrofuran (THF)was the elution solvent. The dead volume between theLS and UV detectors was in the Ðrst arrangement*V \ 0É02 ml and in the second one *V \ 0É01 ml



(LSÈUV) and *V \ 0É4 ml (UVÈVISC). The volumeswere determined with linear standard polystyrenes.The initial concentration of the samples was at most10 mg ml~1.

Molar mass and viscosity determination of

non-fractionated samples

The non-fractionated samples were characterized bystatic light scattering and capillary viscometry. A modi-Ðed, fully computerized SOFICA goniometer (BaurInstrumentenbau, Hausach) was used that was equippedwith a 0É5 mW He/Ne laser. Calibration was made withthe data of Ref. 29. Measurements were performed withÐve concentrations in cylindrical scattering cells of 8mm diameter. All solutions were Ðltered through Milli-pore TeÑon Ðlters of 5 lm pore size depending on thesize of the macromolecules. The molar mass and theradius of gyration were determined by evaluating Zimmdiagrams, or the modiÐed version of Berry diagrams, inthe usual manner.

The viscosities were measured with an automatic vis-cometer (Schott Co., Mainz) that allowed controlledautomatic dilution in the viscometer. A Ubbelohde vis-cometer with a capillary size of 0É4 mm was used. Theinertia correction remained negligibly small.

Some general remarks on SEC with on-line

UV /LALLS /VISC detection

Molar mass distributions of linear macromolecules aremostly determined by common size exclusion chroma-tography using a separately determined calibrationcurve of the molar mass as a function of the elutionM

ivolume This technique cannot be applied tove, i .branched or cyclic macromolecules, since separationtakes place in the column with respect to the hydro-dynamic volume of the samples. Obviously a branchedmacromolecule of a certain hydrodynamic volume has amuch higher mass than the corresponding linear chainof the same chemical composition.

To get reliable results on the molar mass, the SECdevice was equipped (in addition to the UV detector)with a LALLS detector in one series of measurementsand with LALLS and VISC detectors in the other. Thisset-up gave the signals

detector),I SiP c

i(UV

detector),I SiP c

iM

i(LALLS

detector).I SiP c

i[g]

i(VISC

The combination of these signals allowed the computa-tion of

as a function ofI ci

Mi,

as a function ofI [g]i

Mi,

where is the mass fraction of polymers in the exclu-ci

sion volume element and and are the corre-ve, i , Mi

[g]i

sponding molar mass and intrinsic viscosity number.Often, however, the “calibrationÏ function, i.e. the curveof as a function of exhibits strong ÑuctuationsM

ive, i ,

at large elution volumes. This is a consequence of thefact that and are small in this region and are nec-c

iM

iessarily associated with large errors. A typical exampleis shown in Fig. 1 (two columns in series). To avoid thisuncontrolled noise, the calibration curve was linearlyextended to larger values of This will be justiÐedve .only up to a region where separation of the gel matrixused is still e†ective. A test with linear PS chainsrevealed that linearity is obtained up to ml forve \ 19É2the two combined columns and ml for theve \ 9É6linear separation gel. These values correspond to molarmasses M \ 28 000 and 15 000 respectively. Since themolar mass of the unimer, i.e. the star molecule, is

the applied correction appears justiÐed. AM1 w \ 60 000,separately determined calibration curve with PS stan-dards and with four three-arm star molecules of di†er-ent arm lengths gave very similar results (Fig. 2). This isevidently a consequence of the fact that three-arm star

Fig. 1. UVÈelution diagram (full line) and molar mass versuselution volume calibration curves : raw calibration curve ;|,

linearized calibration curve. Measurements were made with>,a crosslinked sample of g mol~1 with a columnM1 w \ 318 000

combination of 104] 106 in THF.Ó

Fig. 2. Calibration curve determined by linear PS standardsand three-arm PS star molecules of various arm lengths.



molecules deviate only weakly in their hydrodynamicbehaviour from linear chains.30

RESULTS

Molecular polydispersity

A Ðrst characterization of a molar mass distribution isgiven by the polydispersity parameter which,M1 w/M1 n ,as already pointed out, is related to the width p of thedistribution as

M1 wM1 n

\ 1 ] p2M1 n2

(13)

In contrast with linear chains, this polydispersity ratiodoes not remain approximately constant for randomlybranched samples, but increases immensely with theweight-average molar mass.9,16 Figure 3 represents theexperimental results obtained with 25 samples of di†er-ent (open symbols), where the two molar mass aver-M1 wages were evaluated by the well-known relationships

M1 w \ ;iciM

i;

ici

, M1 n\ ;ici

;i(c

i/M

i)

(14)

The full line gives the theoretically predicted curve thatwas calculated for polycondensates by the tech-A3B2nique of cascade theory,15 where represents theA3three-arm star molecule with g mol~1 andMA \ 60 000

the diisocyanate with g mol~1, and whereB2 MB \ 184full conversion of the isocyanate groups was assumed.The relationships for the corresponding molar mass

Fig. 3. Molar mass dependence of polydispersity ratio M1 w/M1 nof 21 samples from pre-gel state and four from sol fraction ofpost-gel state. The dotted line represents the curve accordingto the Stockmayer distribution ; there is no di†erence for per-

colation predictions.

averages are given in the Appendix. Good agreementbetween theory and experiment is observed, which hasthe following basis. The number-average molar mass issolely determined by stoichiometry, and remainsM1 nÐnite even at the gel point, where grows beyond allM1 wlimits. Thus in any theory, must grow asymp-M1 w/M1 ntotically proportional to According to Fig. 3, thisM1 w .asymptote is not fully reached at or aM1 w/M1 n \ 6degree of crosslinking X 4M1 w/M1 star\ 20.

Raw and corrected molar mass distributions

Using the individual calibration functions for eachsample, as shown with an example in Fig. 1, one obtainsmolar mass distributions as shown in Fig. 4 for ninedi†erent samples from the pre-gel region. Very similarcurves were obtained with the three samples of the solfraction from the post-gel region. For all samples fromthe pre-gel state a double peak in the distribution wasobserved at a molar mass slightly less than M

i\ 80 000,

but for the samples from the post-gel state only a mono-modal distribution was obtained. This e†ect is discussedlater. Before entering the discussion, we have to appre-ciate that the “distributionsÏ in Fig. 4 are actually onlyraw distributions or, more precisely, histograms. Thisfact becomes clear on recalling the general dependenceof a linear calibration curve :

ve, i\ K1[ K2 ln(Mi) (15)

For equally sized of the elution volumes one*ve, iobtains an average over the molar mass proportional to

but at small elution volumes a much larger massln(Mi),

fraction is detected than at large elution volumes,because the molar mass interval over which the averageis collected is logarithmically spaced, i.e. M

i\ [exp(K1

which introduces a serious error.[ ve, i)]1@Ki,This error can be corrected by forming Ðrst the inte-

gral or cumulative distribution followed by aW (Mj)

Fig. 4. Raw molar mass distributions (histograms) of pre-gelsample 4@ g mol~1)(M1 w, LS\ 235 000



Fig. 5. E†ect of transforming raw molar mass distribution (histogram) into true molar mass distribution, demonstrated withsample of g mol~1. For details see text.M1 w, LS\ 235 000

Fig. 6. Fits of experimental distribution curves with Stockmayer distribution (mean Ðeld approximation) for four pre-gel samplesand one from post-gel state : X \ 5É1 (post) ; X \ 5É3 ; X \ 11É1 ; X \ 22 ; X \ 72. Each curve is shifted by two|, K, L, ), È,decades to avoid collapse of the curves. The slope of the curve for X \ 22 is 1[ q\ [1É51 (mean Ðeld : and()) 1 [ qFS\ [1É50)for X \ 72 it is 1[ q\ [1É36 (percolation : where Full-line Ðts by Eqn (8@) ; broken-line Ðts(È) 1 [ qperc\ [1É2), X \M1 w/M1 star .

by Eqn (4@) or procedures as indicated on the Ðgure.



numerical di†erentiation at the position Mi:

W (Mj) \ ;

i

jm(M

i)

w(Mi) \AL[W (M

j)]

LMj

Bat Mj/Mi

(16)

where the mass fractions have been normalized suchthat

;all i

m(Mi) \ 1

The result of this correction is seen in Fig. 5 for oneexample. A much narrower distribution is nowobtained, but it is burdened with a large error in thelow-molar-mass region because of the numerical di†er-entiation manoeuvre.

Fit of the distribution functions w(Mi)

It is of interest to check how well the corrected molarmass distributions are described by the Stockmayer dis-tribution, which for f \ 3 reduces to

w(Mi) \ d0

(1 [ a)2a

3(2n

i) !

(ni[ 1) !(n

i] 2) !

] [a(1 [ a)]ni (4@)Here the degree of crosslinking of the individual clustersis which has to be distinguished from X 4n

i\ jM

i,

used to describe the degree of crosslinking ofM1 w/M1 starthe non-fractionated samples. may be takenj B 1/M1 staras a Ðt parameter. Alternatively, for large one hasn

ifrom Eqn (8)

w(Mi)\ c0

1 [ aa)n

[4a(1 [ a)]jMi`1(jM

i] 1)3@2 (8@)

Additional Ðt parameters and have been intro-c0 d0duced in these two equations, since for experimentalreasons and because of the approximation applied inderiving Eqn (8), the various distribution functions arenot uniquely normalized. In Fig. 6 the best Ðts by Eqns(4@) and (8@) are demonstrated with four examples fromthe pre-gel state and one from the sol fraction of thegels. The degree of crosslinking changes in these exam-ples from X \ 5É3 to 72 and is X \ 5É1 for the post-gelfraction. The Ðt with the approximation of Eqn (8@)was made with a Marquardt programme settingj \ 1É67 ] 10~5 mol g~1 in one series (two-parameterÐt : and a) and with j as an additional Ðt parameterc0(three-parameter Ðt : a and j). These results werec0 ,compared with the two-parameter Ðt of Eqn (4@)

DISCUSSION

Precursor fraction

The double peak that occurred in the distribution func-tions of Fig. 4 at low molar masses is already seen in the

elution diagrams of as a function of the elutionci

volume. Figure 7 gives an example. Since in most casesthe degree of crosslinking does not exceed a value of 10,it is reasonable to presume that the non-reacted starfraction causes the peak around ml, while theveB 18other part represents the crosslinked fraction. Figure 7shows an attempt to separate these two fractions. Thisseparation cannot be made by common programs suchas PEAKFIT, since there the type of distribution needsto be known. However, since the distribution of the starmolecules could be separately determined with the samecolumn combination, it became possible to split themeasured elution curve into the fraction arising fromthe star precursor, and the part of the crosslinkedmst ,stars, This procedure was applied to 17 samplesmbr .and the result is plotted as versus in Fig. 8. ThisM1 w mbrbranched fraction is related to the extent of cross-linking. Since the molar mass of the diisocyanate is

Fig. 7. Separation of measured elution diagram into fractionsof non-reacted star and crosslinked molecules (M1 w \ 319 000g mol~1). The curve denoted “3-arm star molecule distribu-tionÏ is the elution diagram of the star multiplied by the frac-

tion of free star molecules.mst\ 1

Fig. 8. Molar masses of crosslinked fractions against massfraction of crosslinked fraction. Divergence of is predictedM1 w

for mbr, c \ 1 [ (1 [ ac)3 \ 0É875.



small compared with the molar mass of the star andsince all isocyanates have fully reacted, the crosslinkingreaction can be well approximated by a simple tri-functional homo-polycondensation (A3-typepolycondensation). In the FloryÈStockmayer schemeone then has

mbr \ 1 [ mst , mst\ (1 [ a)3, a \ 1 [ (mst)1@3 (17)

Table 1 gives a list of the data according to Eqns (17).Evidently a point of gelation is approached where M1 wincreases beyond all limits. This critical behaviour isdiscussed in separate papers in the context of other pro-cedures for determining the gel point.27,28,30,31 Inter-estingly, the elution diagrams of the samples from thesol fraction of the post-gel state are monomodal. This isindeed to be expected, as the mass fraction of non-mstreacted stars will be very small in that region of highconversion.

Shape of the distributions

Two facts are immediately seen. (i) The approximationof Eqn (8@) agrees exceedingly well with the exact dis-

tribution of Eqn (4@) when and even atni\ jM

i¹ 5,

lower values of the agreement is good despite theMi

scatter of experimental data. For this reason, only thetwo- and three-parameter Ðts of Eqn (8@) were appliedfor the samples with (ii) DeviationsX \M1 w/M1 star [ 6.from StockmayerÏs distribution become noticeable whenX [ 10 ; the experimental curve decays faster at large

than predicted and indicates a stronger-decayingMi

cut-o† function than given by Eqn (12a). Of course, abetter Ðt could be made with a three-parameter Ðt, butthen the molar mass of the star molecules would haveto be assumed as a varying parameter, which is notreally sensible. Finally, near the gel point, i.e. X [ 22,the experimental curve could be well described by anexpected power law. The line in Fig. 6 corresponds toan exponent of which lies just in1 [ qexp\ [1É36,between the predicted exponents of the mean Ðeldapproximation and percolation(1[ qMF\ [1É5)theory (1 [ qperc \[1É2).

Crossover from mean field to percolation regime

Since (to be considered as a normalizationc0parameter) was chosen as a Ðtting parameter, the

TABLE 1. Molecular averages obtained by SEC/LALLS compared

with direct measurements by static light scattering. The first 19

samples were obtained with the combination of two styragel

columns (104 + 105 and from the pre-gel state. The last fourA� )samples are from the post-gel and were obtained with the linear

styragel column. All molar masses are in 105 g mol—1. denotesmbr

the branched fraction without the fraction of non-reacted stars

Sample M1w, SEC

M1n, SEC

M1w/M1

nM1

zM1

w, LSm

br

1 pre 2·01 1·11 1·81 3·92 2·6 0·490

2 pre 2·16 1·21 1·78 4·22 3·28 0·545

3 pre 2·43 1·28 1·90 8·00 2·30 0·531

4 pre 2·41 1·30 1·85 4·85 2·35 0·547

5 pre 3·03 1·19 2·54 6·76 4·40 0·642

6 pre 3·19 1·39 2·29 8·98 3·45 0·666

7 pre 3·38 1·35 2·50 7·13 3·80 0·658

8 pre 3·59 1·39 2·58 8·69 3·65 0·682

9 pre 3·60 1·56 2·31 34·4 4·50 —

10 pre 4·14 1·62 2·55 13·8 4·80 0·681

11 pre 4·59 1·60 2·87 23·8 5·10 0·706

12 pre 5·07 1·41 3·59 22·2 5·65 0·712

13 pre 5·20 1·86 2·79 17·7 4·78 0·734

14 pre 6·19 1·94 3·19 1·98 5·70 0·740

15 pre 6·64 2·32 2·86 18·1 6·80 —

16 pre 9·19 1·58 5·82 3·18 8·00 0·767

17 pre 9·29 3·21 2·89 30·7 7·80 0·806

18 pre 9·63 2·14 4·50 42·6 8·65 —

19 pre 12·5 2·88 4·34 3·95 9·25 0·818

20 pre 12·9 1·78 7·25 65·7 21·0 —

21 pre 43·0 4·74 9·07 206 56·5 0·861

22 post 6·72 1·49 4.51 47·8 — —

23 post 6·42 1·83 3.53 20·5 5·50 —

24 post 3·04 1·39 2.18 7·72 — —

25 post 1·97 0·97 2.03 6·21 — —



resulting a values cannot be expected to have the orig-inal meaning of crosslinking probabilities. None theless, using the approximation of Eqn (6) with M0\

one obtains the data as given in TableM1 star , M1 w, FS(afit)2 compared with obtained directly by SEC/M1 w, SECLALLS. This comparison demonstrates that the FSapproach is a good approximation (within 10%) for thereal system up to a degree of crosslinking X \ 5, andeven at X \ 10 the deviations are no larger than 20%.At X [ 5 the FS approach becomes increasingly poorand eventually invalid when the gel point isapproached. Hence we can state that the region ofcrossover to percolation starts around X \ 5.

The other limit of the crossover region where perco-lation becomes valid could not be determined, since inthis region the molar mass is already too high for areliable molar mass determination and a suitable SECrun. Figure 9 shows how the expected power-lawbehaviour develops when the degree of crosslinking isincreased. First indications are obtained for X [ 10,while for X [ 70 the power-law behaviour appears tobe fully developed. The slope gives a critical exponentq\ 2É36, which is lower than predicted from perco-lation. It appears conceivable that the critical region hasnot yet been reached. A realistic estimate for the onsetof true percolation behaviour may be at X [ 200.

TABLE 2. Comparison of molar masses calculated fromM1w, FS

(afi t

) afi t

,

i.e. Eqn (6) with obtained from SEC measurements.M1w, SEC

X =is the degree of crosslinking, where AllM1

w, SEC/M1

starM1

star= 60 000.

samples are from the pre-gel state. Note that the numbering of the

samples is not the same as in Table 1

Sample afit

M1w, FS

(afit

) M1w, SEC

M1w, FS

(afit

)/M1w, SEC

X

1¾ 0·288 1·98 2.01 0·99 3·35

2¾ 0·305 2·18 2.16 1·01 3·60

3¾ 0·319 2·38 2.43 0·98 4·05

4¾ 0·379 3·72 3.59 1·04 5·98

5¾ 0·375 3·59 3.19 1·13 5·32

6¾ 0·389 4·08 3.38 1·21 5·63

7¾ 0·384 3·89 4.14 0·94 6·90

8¾ 0·416 5·50 5.20 1·06 8·67

9¾ 0·440 7·93 6.19 1·28 10·3

10¾ 0·433 6·97 6.64 1·05 11·1

11¾ 0·456 10·1 9.63 1·05 16·1

12¾ 0·479 22·9 12.5 1·83 20·8

13¾ 0·493 69 12.9 5·4 21·5

14¾ 0·499 499 43.0 11·6 71·7

Fig. 9. Change in shape of mass fraction molar mass distribution with increasing Power-law behaviour starts at aboutw(Mi) M1 w .

X \ 22, but the slope decreases further with increasing X until X [ 72 is reached.



Fig. 10. Dependence of on maximum molar mass in LSÈM1 welution diagram The slope of the curve is(M1 max B M1

z). 3giving[ qexp \ 0É64 ^ 0É05, qexp\ 2É36 ^ 0É05.

Another way of determining the exponent q resultsfrom the relationship

M1 w P M1z3~q (18)

which is obtained after calculating the required averagesfor and with the distributionM1 w M1

z

w(Mi)\ AMi1~qf

AA Mi

1 [ p/pc

BpB(11@)

in which p and are the non-critical and critical prob-pcabilities of crosslinking, or lattice site occupation, in theFS theory and lattice percolation theory respectively.The parameter p is another critical exponent of the dis-tribution that deÐnes the z-average molar mass andwhose value again di†ers in the two theories.21 Its mag-nitude will not be considered further in this contribu-tion but will be discussed elsewhere in the context of theother critical exponents.26,28 The elution diagrams ofthe light-scattering intensities always pass through apronounced maximum that results from the fact thatthe scattering intensity is It has beenRh, iP c

iM

i.

estimated25 that

Rh, max 7 M1 maxP M1z

(19)

Thus a double-logarithmic plot of againstM1 w M1 maxshould result in a straight line with an exponent of3 [ q. Figure 10 shows that this plot has a slope of3 [ q\ 0É64 ^ 0É05, which gives q\ 2É36 ^ 0É05.

CONCLUSIONS

In the present study the molar mass of the repeatingunit in a random crosslinking system was chosen to bethat of a fairly large three-arm star molecule. By thischoice it became possible to make careful measurementswith samples of low degrees of crosslinking. These mea-surements allowed reliable estimations in a regionwhere the mean Ðeld approach can be applied safely

and where the crossover to percolation occurs. Up to adegree of crosslinking the FS theoryX \M1 w/M1 0\ 5gives excellent agreement with experiments. Even atX \ 10 the deviation between the FS distribution andthe experimental Ðndings is less than 10%. At largervalues, signiÐcant deviations occur, and at X B 70 aclear power-law behavior is obtained. The exponent(q\ 2É36 ^ 0É05), however, is smaller than ofqFS\ 2É5the FS theory but still larger than the percolation pre-diction Possibly even at X \ 70 the trueqperc\ 2É20.percolation limit has not yet been reached, and the criti-cal region of gelation may not be valid until X [ 200.

ACKNOWLEDGEMENTS

We are grateful to Professor W. H. Stockmayer, Dart-mouth College, Hanover, NH, USA, who followed witheager interest the progress in this study. His numerousfruitful comments were helpful to us and a source ofgreat pleasure. The present work was kindly supportedby the Deutsche Forschungsgemeinschaft within thespecial research area SFB 60.

APPENDIX

Random branching processes can be most efficientlydescribed by the cascade theory that is based on ageneral scheme developed by Good10 and was adaptedto polymer science by Gordon.11 Using vectorÈmatrixnotation, the Markovian branching process of copoly-mers can be expressed in a very condensed form.15 Herewe quote only the three quantities gel point

o 1 [ P o\ 0 (20)

weight-average molar mass

M1 w \AmAmB

BCAMAMB

B] N1(1 [ P)~1

AMAMB

BD(21)

and number-average molar mass

M1 n\

AnAnB

BAMAMB

B

1 [ 0É5CAnA

nB

BA fa2bBD (22)

where and are the mole fractions of the star andnA nBthe isocyanate respectively and and are the corre-mA mBsponding mass fractions, which are related to the molarmasses and asMA MB

mA \ 2bMA2bMA ] f aMB

, mB\ faMB2bMA ] f aMB

(23)

Here is the population matrix whose elements giveN1the number of A and B units in the Ðrst generation andP is the probability transition matrix that gives theprobability for an A unit in the nth generation being



connected to another A or B unit in the (n ] 1)th gener-ation and similarly for a B unit in the nth generation. Inthe present case these two matrices are given by

N1\A 02b

f a0B

, P\A0b

( f [ 1)a0

B(24)

The matrix products in Eqns (21) and (22) can be per-formed analytically with the result

ac \ [bc( f[ 1)]~1(gel point, Flory condition)

(25)

M1 n \ nA MA ] nBMB1 [ 0É5( f anA ] 2bnB)

(26)

M1 w \ (T1] T2)/T3 (27)

with

T1\ mA[(1] ab)MA ] a f MB]

T2\ mB[2bMA ] (1 ] ab)MB] (28)

T3\ 1 [ ab( f[ 1)

The relationships used in the text result for b \ 1 (fullisocyanate conversion).

REFERENCES

1 (a) Flory, P. J., J. Am. Chem. Soc., 58 (1936) 1877 ; (b) Principles ofPolymer Chemistry, Cornell University Press, Ithaca, NY, 1953.

2 Schulz, G. V., Z. Phys. Chem. B, 43 (1939) 25.3 Hibbert, H. & Perry, S. Z., Can. J. Res., 8 (1933) 102.4 Flory, P. J., J. Am. Chem. Soc., 62 (1940) 1561.5 (a) Szwarc, M., in Encyclopedia of Polymer Science and T ech-

nology, Vol. 8, pp 303È325, eds H. F. Mark, N. C. Gaylord &N. M. Bikales. Wiley, New York, 1968 ; (b) Henderson, J. F. &Szwarc, M., Macromol. Rev., 3 (1968) 312.

6 Bamford, C. H., Barb, W. G., Jenkins, A. D. & Onyon, P. F., T heKinetics of V inyl Polymerization by Radical Polymerization.Butterworths, London, 1958, pp. 283È290.

7 Peebles, L. H., Molecular W eight Distributions in Polymers. Wiley,New York, 1971.

8 Feller, W., Probability T heory and Its Applications, 3rd edn. Wiley,New York, 1968.

9 Stockmayer, W. H., J. Chem. Phys., 11 (1943) 45.10 Good, I. J., Proc. Camb. Philos. Soc., 45 (1940) 360.11 (a) Gordon, M., Proc. R. Soc. L ond. A, 268 (1962) 240 ; (b) Gordon,

M., Malcolm, G. N. & Butler, D. S., Proc. R. Soc. L ond. A, 295(1966) 29.

12 Miller, D. R. & Macosko, C. W., Macromolecules, 9 (1976) 206 ; 11(1978) 656.

13 Macosko, C. W. & Miller, D. R., Macromolecules, 9 (1976) 199.14 Kajiwara, K., Burchard, W. & Gordon, M., Br. Polym. J., 2 (1970)

110.15 Burchard, W., Adv. Polym. Sci., 48 (1983) 1.16 Flory, P. J., J. Am. Chem. Soc., 61 (1939) 3334 ; 62 (1940) 2261.17 Jacobson, H., Beckmann, C. O. & Stockmayer, W. H., J. Chem.

Phys., 18 (1950) 1607.18 Stockmayer, W. H. & Weil, L. L., in Advancing Fronts in Chem-

istry, ed. S. B. Twiss. Rheinhold, New York, 1945.19 Stanley, H. E., Introduction to Phase T ransitions and Critical Phe-

nomena. Clarendon, Oxford, 1971.20 Isaacson, J. & Lubensky, T. C., J. Physique, 42 (1981) 175.21 Stau†er, D., Introduction to Percolation T heory. Taylor & Francis,

San Francisco, CA, 1985.22 de Gennes, P.-G., Scaling Concepts in Polymer Physics. Cornell

University Press, Ithaca, NY, 1979.23 Daoud, M. & Martin, J. E., in T he Fractal Approach to heter-

ogeneous Chemistry, ed. D. Avnir. pp 109È130. Wiley, New York,1989.

24 Zimm, B. H. & Stockmayer, W. H., J. Chem. Phys., 17 (1949) 1301.25 Schossler, F., Benoit, H., Grubisiv-Gallot, Z. Strazielle, C. &

Leibler, L., Macromolecules, 22 (1989) 400.26 Weissmu� ller, M. & Burchard, W., Acta Polym. (1997) acepted.27 Weissmu� ller, M. & Burchard, W., Macromol. Chem. Phys., sub-

mitted.28 Weissmu� ller, M., PhD T hesis, University of Freiburg, 1996.29 Bantle, S., Schmidt, M. & Burchard, W., Macromolecules, 15 (1982)

1604.30 Stockmayer, W. H. & Fixman, M., Ann. NY Acad. Sci., 57 (1953)

334.31 Weissmu� ller, M., Merkle, G. & Burchard, W., Macromol. Symp., 93

(1995) 301.


molar mass distributions of end-linked polystyrene star macromolecules

Documents