Conformation of Polymer Chain in the Bulk

Conformation of Polymer Chain in the Bulk

J.P. Cotton,1®D._Decker,11·H.

Benoit,lb

^B.Farnoux,1®J. Higgins,1'^G.Jannink,1®

R.Ober,ld^C.Picot,lb and J.^desCloizeaux1®

Centre de Recherchessur lesMacromolécules (CNRS),⁶⁷⁰⁸³StrasbourgCedex,France;Servicede Physique^duSolideetdeResonanceMagnetique and^ServicedePhysique Théorique,^Centre d’EtudesNucléairesde Saclay, 91190Gif-sur-Yvette, France; Instituí Max^VonLaue-Paul Langevin,38042 Grenoble,France; and Laboratoire^delaMatiere Condensée, College deFrance, 11,PlaceM. Berthelot,75005Paris,France. ReceivedMay 2,1974

ABSTRACT: Neutron coherent scatteringtechniques have been^usedfor thedetermination of the conformation of polymer in bulk^andexperimental details^are givenabout the application of this method^tothe study of polymeric systems. Measurements have been madefor smalland intermediatemomentum ranges ^{on a}seriesof eight^mono- dispersedeuterated polystyrenes of molecular weight ranging from21,000to 1,100,000.Theresults lead tothecon-

clusonthatinamorphous state theconformationofthepolymer^moleculeisindistinguishable fromthatin solvent andthatthe Debyescattering function which^isvalid for unperturbedchains appliesforq^{~1 as}lowas 10Á.

Inorder to explain^suchproperties^as rubberelasticity it has long ^{been assumed} that polymer chains in the liquid state obey Gaussian statistics.2,3Below its _glass transition temperature ^a polymer ^becomes hard and loses its elastic properties. It is believed,4however, that this transitionaf- fectsonly^themotion ofthe drawn_segments and not their configuration^whichisexpected^to remainGaussian.There is much indirect evidence in favor of this _hypothesis but until recently it has been impossible ^to prove it directly.

Moreover, many ^authors5,6believefrom theoreticalconsid- erations, electron microscopy, and diffraction results that thereismore orderinthe amorphous state,takingtheform ofsupramolecular structures.

In this_paper, we present the resultsofmeasurements of the conformation of polystyrene molecules in the bulk amorphous ^state. These measurements were made using theneutron small_anglescattering apparatus^attheInstitut Laue-LangevininGrenoble and attheCEA_Saclay.Similar experiments in this field have already ^been published by Kirste7 and Ballard.8

Theneutron SAStechnique^isparticularlywell suitedto the problemfortwo reasons.

(1) Neutronwavelengths availablefromthe coldsources at each ofthe installations are ofthe order of 10Á. It is thus possible to match the experimental ^wave vectors to the molecular dimensions. In

light

scattering experiments only radii of gyration of very largemolecules can be mea-

sured, whileX-rayexperiments^need extremely small^scat- teringangles.

(2) The difference between the neutron scattering length of deuterium and hydrogen ^is particularly ^useful since it will be shownthat deuteration ^does not affect the thermodynamic properties ofthe chains and that there is perfectcompatibilitybetween deuterated and hydrogenous molecules. Thus, if deuterated molecules are embedded in

a matrix of hydrogenous molecules, the scatteringpattern ofthe formermay be measured and the molecular dimen- sionsandthe statisticaldistributionof the chain_segments deduced. The contrast is large enoughfor_very lowconcen-

tration of_{polymer to}beused.

In the first part ofthe paper ^we givedetailsoftheneu-

tron technique ^as applied to the problem of_polymer con-

formation, then the samplepreparation^isdescribed and fi- nally^theresults andtheir interpretation^{are discussed.}

Neutron Small Angle

Scattering

Technique

For experiments ^on polymeric systems two principal propertiesdifferentiateneutron scatteringfrom other tech- niques: the interaction with the sample depends only ^on the neutron-nucleus interaction _leading to results de-

scribed in part A ofthis section; the wavelengths available rangefrom 1 to 12Á giving^values ofthe scattering vector which are inaccessible to

light

^and X-ray scattering. The apparatus^isdescribedin part^B.

(A)

Scattering Intensity.

^The intensity scattered by systems with long-range correlations (greater than ¹⁰ Á) takes the form ofa central peak about the forward direc- tion. The amplitude of this peak varies stronglyfrom ^nu- cleus to nucleus. Isotopicsubstitutionusedin neutron scat- tering gives ^a powerful method of labeling molecules ^be-

cause it leavesthe chemical properties ofthe moleculesun-

changed.

Wenow evaluate the scattering^cross sectionofamixture oflabeledandunlabeled polymer molecules with a viewto its application ^to deuterated and nondeuterated polysty-

rene. Using the relation for the scattered intensity ^which

we derived,^we considerthechoiceof_sample.

(1)

Scattering

^{CrossSection.The}neutron-nuclear in- teraction is characterized by ^a scattering length o¿ which takes into account the spin ofthe isotope of the nucleus considered. The scattering ^cross section _a(q)perunit solid angleofa neutron of_wavelengthX scattered at an angle by^a monoatomic system consistingof Natoms at positions r¡ iswritten9

v(q) ⁼

Z^(ei9irrri))

⁽¹⁾

il

q isthe scattering vector |q| ⁼ (4ir/X) sin 6/2, the barover

the scattering amplitudes ^{denotes a} spin and ^an isotopic averaging while the broken brackets indicatea thermalav-

erage of the function inside. a(q) may ^be written in the form

a{q) ⁼

a2J^(ei“iri-rJ>)

⁺

Na2

₍₂₎

ij

This relationshipintroduces the coherent scatteringampli- tude ^a and the incoherent scattering ^cross section _a\ de- fined_by^{a =} a¡ and by

⁼

4 2

⁼

4 ( ^

^- a2) (3) andforwhich tablesofexperimentalvaluesexist.10,11Thus the scatteredintensitydecomposesinto ^a coherentintensi-

ty

depending^on the scattering^vector andan incoherentin- tensity

2.

^{Values ofa and are} given in Table I. It is evidentfromthetablethat thevaluesof the scattering^am- plitudes ofhydrogen anddeuteriumare very different.The transmission of _{the system} T ⁼

I(x)/I(0)

^is given by the equation

T ⁼

exp[-d(4na2

⁺

) ]

⁽⁴⁾

Downloaded via UNIV OF CINCINNATI on January 22, 2021 at 00:26:54 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

where disthe densityofthe_{system and} x the thicknessof the_sample.

Expression^{2takesno account} of multiplescattering and

can only^{be used when} T is ofthe order of unity. Let^us consider a system of N deuterated molecules each com-

posedof^m atomsofwhichthe scattering amplitudesare aQ

and ai„ (avaryingfrom 1to m). If ^thevalues ofthescat- tering ^vector q ^are such that _q~x is large relativeto the molecular dimensions, the structure factor ofthe molecule may ^beneglected (this ^isin factalways true forsmallscat- tering angles). In this case, it is possible to generalize to molecules the idea of coherent and incoherent scattering amplitudes A ^andA\suchthat

m

A ⁼ _aa Aj2 ⁼ aia2 (5)

o=l o=l

and thecross sectionthenbecomes

a(q) ⁼ A2S(q) ⁺ NA,2 ⁽⁶⁾

whereS(q)^isthe densityfluctuationcorrelationfunction of thecentersofmass ofthe molecules

S(q) ⁼ (eig(Ri-Rt>) (7) ij

Usingexpression⁵andTable Ithevaluesofthe scattering amplitudes<zhand am ofa monomer C8H8ofhydrogenous polystyrene ^{and an} and am of a monomer of deuterated polystyrene may^becalculated.

aH ⁼

2.344

x 10"12 cm am^{2 =}

50.7

xlO'24 cm2 aD ⁼ 10.672 ^{x 10'12 cm} am2 ⁼

1.46

xlO'24 cm2 (8) Consider a system consisting of a mixture of IVa mole- culesof_{type A}withcross sectionsA and Ai and

Nb

^mole-

culesof_type B with ^cross sections B andB_i ofthe same

volume. Using only the hypothesisoftheincompressibility ofthemixture^we can writethe scattering^cross sectionin theform12

a(q) ⁼ (A ^- B)2S(q) ⁺ N^A,2 ⁺

NbB2

⁽⁹⁾ which demonstrates the contrast factorK2 ⁼ (A ^— B)2 be- tween the two typesofmolecules. S(q) ^istheFourier trans- form ofthecorrelation function ofthe A molecules in the presenceof_{B, or,} which is the^same thing, of^holes in the ensemble B causedby the presence ofA. Thuseven if the coherent cross section ofA is zero the scattered intensity givesthe correlationbetweenthe Amolecules.

(2) Signal

Intensity.

^{When the} sample ^isa mixture of polymers ofmolecular weightMlabeled by isotopic substi- tution (A)andunlabeled polymers (B) the scatteredinten- sitymay^bederivedfrom_{eq 9}in theform12

I(q)

⁼

7 0

^K2m* i MSAq) ⁺ _m,

A,2

⁺

^ B2'1+ _{f* N(t,q)dt}

(10)

mB j

where N isAvogadro’s Number, V the volume ofthe_sys- tem incm3,

^{theincidentneutron flux in}cm_2sec_1,^tthe counting time in seconds, T the transmission _(eq4) of the system. Ca and Cg ^are the concentrations in _g cm-3, m&

andmg ^are the molecular weights ofthemonomers A and B, and K2,A\,^andB\^are the valuesof the contrast inscat- teringamplitude for the labeledandunlabeledmonomers.

Table I

Atom a x 10"12cm ^{x 10'24cm2}

H

-0.374

79.7

D 0.667 7.2

C 0.665 0.01

Ref 11 9

S(q) ^is the Fourier transform of the correlation function betweenallthe markedmolecules,normalized tounity^atq

= 0.N(t,q)^isthe instantaneousrandomnoiseof_{the appa-} ratus. We can now discuss the amplitude ofthe scattered intensityand the signal^{to noise}ratiofor thetwo cases.

Thesurface area ofthe _sample isdetermined by the di- mensions of the incident beam and of the counter. Its thickness issuch that thetransmission issufficientlyhigh to reduce themultiple scattering. Equations ^{4 and 8show} that at constant transmission^a sample ^{can be}ten timesas

thick forthe deuteratedmatrixasforthe hydrogenous^ma- trix. However, the multiple scattering arising from an H matrixcomes from the strong incoherent scatteringampli- tudeofhydrogen^andistherefore independentof_scattering vector. Thisis not true in thecase ofaDmatrixwherethe multiplescattering effects ^arisefromthe coherent scatter- ing^andare peakedin theforward direction.

The signal^{to noise}ratioS/N, neglecting the^noiseofthe apparatusitself,^is

_

g*S„(g)(M/mA)

Aj2 ^+· (Cb>?Za/Ca^b)-®i

Asan example^{we have}calculated the tworatiosfora solu- tion ofa polymer ofmolecular weight ^{10® at a concentra-} tion of^1%in the very lowqregion.^Thesecalculationsindi- catefirst that^we haveausableS/N ratio inboth situations.

Table II is stronglyin favor ofa deuteratedmatrix. How- ever, thenoise oftheinstrument itself (N(t, q)) ^isoftenof thesame orderas the incoherent background of^aPSHma-

trix and in practice the S/N ratio for ^a D matrix is only threeor four times betterthanforan H matrix.

Moreover, any density fluctuations or impurities ^{in the} samplewill_producea coherent signal which^isvery intense for the deuterated matrix due to its _large coherent cross

section and ispeaked around the forward direction unlike the incoherent background from the hydrogenous matrix whichisisotropic. ^Suchasignal^{was observed}from all

fully

deuterated polymer samples and proved very difficult to remove fromthe results. Thus in the smallq rangethe^use of^a deuterated matrixisnot obvious andwe found an hy- drogenous matrix more satisfactory. In _discussing the in- termediateqrange^we showthatthesame arguments hold.

(B) Apparatus. Inorder to study the characteristicpair correlationfunction of_{polymersamples}^we require scatter- ingvectors (q ⁼ (2irsin

)/ )

^{with valuesbetween10~3and}

10_1A-1. Toreach these valuesthe experimental^arrange- ment must have very small scatteringangles

(

^~ Io) ^anda long wavelength (greater than ⁴ Á). Furthermore in order to explore^thesetwo limits of_{the range}ofqwhich_givepre- cise informationon chain statistics,^as will bemade precise in the discussion, two types of experimental arrangement

are necessary andtheywillbedescribed separately.

(1) General Points about the Apparatus. ^Continuous neutron beams ^come from nuclear reactors in which neu-

trons are produced by nuclear fission ^at thermal equilibri-

um with themoderator at a temperature ofabout 373°K.

These neutrons are classified according ^to their energy E (in eV)relatedto themoderator temperature T (in^degrees

Table II

Marked

Polymer

^Matrix S/N

PSD PSH 13

PSH PSD 380

TableIII

Vertical divergence Horizontal divergence

L,

^{m x 10s radians x 103 radians}

40 13 0.8

20 2.5 1.5

10 5 3

absolute) and^to the wavelength(inÁ) by the equation E ⁼

T/l.

^{6 x 104 = 8. 18 X} 10"2/X2 (12) The corresponding wavelength spectrum ^is quasimaxwel- lian with a maximum at about 1.8 Á. In thisspectrum the intensity ofwavelengthsrequired for^smallanglescattering (E < 0.005, > 4 A) ^is weak. The flux may ^{be increased} andthe maximumofthe distribution shifted_{to long}^wave- lengths by locally replacing the moderator by^a coldsource

consisting of^a cell filled with

liquid

hydrogen ^or deuteri-

um.13·14In thisway^afactorofabout 10isgained in theflux of long-wavelength neutrons (“coldneutrons”).

Neutronguides15·16 (usingtotal internal reflectionof the neutrons)^remove theeffectivesource ofneutrons fromthe reactor itself into an adjacent experimental hall. Curved guides act also^{as a}getter for^neutrons ofshort wavelength.

Figure ^1shows the spectrum ofneutron wavelengths ^com- ingfromthe curved guide of the^reactor EL3at Saclay.

Two typesofmonochromators^were usedforthese exper- iments: either mechanical^or crystalline. Thefirst _type isa

helical slot velocity^selector17with a wavelength resolution ofthe orderof

/

^{= 5 X 10_1 whichmust beconsidered}

wheninterpretingthe results.With acrystalline^monochro- mator neutrons ofwavelength ^are reflected according ^to the Bragg ^law.Fora pyroliticgraphite monochromator the maximum_wavelengthis ⁼ 6.71Áandthe resolution

/

= 10-2. For the same wavelength the intensity of ^mono- chromaticneutrons isabout50times weakerthanfora me-

chanicalselector.

The choice betweenthese two systems^isin effecta com-

promise between the resolution and the signalintensity.

(2) Measurements in the Regionq < 1/Rg. Dll

Ap-

paratus. This apparatus18 ^is constructed at the exit of a

neutron guide giving ^a white spectrum with a maximum wavelength at^5Á. The incidentandscattered divergences may ^be modified _by changing the distances between the

source andthe_sampleandbetweenthe sample and the^de- tector andalsothe wavelengthoftheincidentneutrons.

The apparatus^isshownin Figure^2.

(a) Monochromator. Two mechanical monochromators

were usedintheseexperiments.^One(A)gives^awavelength spectrumofwhich thewidthat half-height

/( )

^isofthe

order of 50% (where < > is the mean wavelength of the spectrum). Thishalf-width isconstant for wavelengths ^be- tween 6 and 10 Á. The second monochromator (B) has a

wavelength spread

/( )

^{of8% at a mean} wavelength of

8.6to 9Á.

(b)

Collimation.

By introducing strength ^sections of neutron guide of_{20, 10, 5,} and 3 m, the distance between the effective source ofneutrons (the end of the guide) and the samplemay ^bevaried between2and40m. Thissource

10"110-2 10'3 ev

Figure ^{1. Plot ofthe} logarithm of^theneutron flux in Ávs. the wavelength measured at theexit oftheneutron guide BI of the Saclay reactor EL3.Thecurves, (a) obtainedwiththe coldsource

and (b) without the cold source, show the neutron gain at large wavelengthin_usingacold^source.

Figure^{2. Schemeofthesmall}angle apparatus D ¹¹ oftheI.L.L.

high flux reactor. On the left a curved guide ^leads the neutron beam to themechanical monochromator (1). Then aguide (2) of variable length translatesthe neutron source to adistanceL from the sample (3). Thescattered neutrons are registered^{on a} multi- counter (4)atvariabledistanceD. Intheseexperiments L D

is3 cm wide by^{5 cm} high. The maximumangular divergen-

ces oftheincident beam for differentdistancesL between samplesandsource are given in TableIII.

(c) Sample Holder. The neutron beam falling ^{on the} sample ^is defined by ^a diaphragm adapted in order ^to match the detectorcellswhich are ofdiameter0.8cm. For temperature-dependent measurements ^an automatic ^con- trolled ^{oven was used.} Measurement of _{sample tempera-} ture was made with an accuracy of ±0.2° and regulation

was within ±0.05°. The sample cell for solutions ^{was a} metal framewithquartz windows^{1 or 2mm} in thickness.

(d) Neutron Detector. The neutron detector is a BF3 counter consistingof64 X 64 ⁼ 4096 cells of^{1 cm} X 1 cm

arranged in ^a square. Measurements were made with the multidetector at distances of10and20m from the_sample (this^{distance is}variablewithin ±1 m due to severaldiffer- ent samplepositions which^are available).

In each experiment the ^{neutron source was at} the same

distance from the sample^as the detector. In these experi- ments, ^where the scattering ^is isotropic, ^{cells are} grouped with reference to their distance from the central cell (i.e., thepoint^wherethedirectbeamtransmittedby the sample falls). Theintensity^{in thecellsata distance}from this cen-

tral cell between R ^— Añ and R + AR is averaged and given^{as an} intensity

I(R)

per ^cm2 (AR ⁼ 0.5cm),R varies from7to 35cm in_stepsof1cm.

The scatteringangle corresponding ^{to each} value of R

Table IV

Experimental

Monochro-

setup

( ),

^A D, ^m mator

A ⁴ 20.36 A

B 6.5 20.36 A

C 8.7 20.36 A

D 9.8 20.36 A

E 6.1 9.33 A

F 6.5 18.66 A

G 7.8 18. 66 A

H 8.7 18.66 A

I 9.8 18.66 A

J 8.8 9.33 B

K 8.7 19.33 B

Figure^3.Schemeofthe setup at the endofthe curved guide BI ofthe Saclay reactor EL 3. One can see the cold neutron curved guide^(a),thebiologicalshielding(b), thegraphite monochromator (c),thegraphitefilter _{(d), the}incidentcollimation slit_{(e),the po-} sition (f)of_{the sample}on the axis ofthe spectrometer, the soller slits _(g)forthe scattered neutron analysis, andthe detector_(h) in itsshielding^atascatteringangle .

is given by ⁼ R/D ^{where D is} the sample detector dis- tance.

(e) Definition of Parameters and

Calculation

ofRe- sults.

(i)

^{Values of} the Angles and

Scattering

Vectors.

For determination ofthe radius of _{gyration Rg} measure-

ments must be made in_{the region q} < 1/Rg. In TableIV

we give the configuration of the apparatus ^and the mean

wavelength

( )

^{used forthese}experiments. Also given^are themaximum6maxand minimum _6mmvaluesofthescatter- ingangle.

We were unable to perform experimentswith themulti- detector at 40 ^m from the sample becauseof the effect of gravity^{on the}trajectory of^theneutrons especially^at long wavelengths making corrections for the wavelength spread in thebeamimpossible.

(ii)

Wavelength Corrections. In ^the Guinier _approxi- mation¹⁹ the scattered intensity^iswritten as a function of wavelengths^andoftheincidentneutrons intensity

I(q)

⁼ 1(6,

)

⁼

/0( )[

^-

(27 / )2( 2/3)]

⁽¹³⁾

where _q has beenreplaced by

2 / .

When theradiationis not monochromatic the measuredintensity^istheintegrat-

ed intensity ^for all wavelengths. If i

( )

^{is the} _intensityof theincidentbeamas afunction of wavelength

/„

⁼

fi(\)d\

⁽¹⁴⁾

R ⁼ 7 cm R ⁼ 35 cm

6min Qmin 6max qma_K

x 103rad ^x 103A'1 x103rad x 103 V1

3.4 5.4 17.2 27

3.4 3.3 17.2 16.6

3.4

2.5

17.2 12.4

3.4 2.2 ^{17.2 11}

7.4 7.7 37.2 38.6

3.7 3.⁶ 18.6 18

3.7 3.0 18.6 15

3.7

2.7

18.6 13

3.7

2.4

18.6 12

7.4 5.3 37.2 2. 38

3.6

2.6

18 13

and

~

fxi(X)dX

*0

<X> ⁼ (15)

( )

⁼

Kj[l

^-

^{2 /x)2Rg/3]i(

^{X)dX (16)}

S' II

S 1 ^-

(27T6)2(Rg2/3)(l//0)/-

^(X)dX'_{X2 J} ⁽¹⁷⁾

we define_I/Xq2 ⁼ /X-2i(X)dX//o^{asthemean yalue}ofX ²of the incidentbeam and_qo ⁼

2 / 0.

The scatteredintensity in thedirection isthenwrittenin the form

( )

⁼ I(q0) ⁼

(

^{1 - (qQ2R2/3)) (18)}

For the wavelengthspread given by the ^two monochroma- torsusedinour experiment,^we obtained

monochromator (A)

⁼

0.95( )

monochromator (B) 0⁼

0.99( )

(3) Measurements in the Region^{q »}

l/R

g.Forvalues of the scattering^vector q »

l/R

gmeasurements are made

on atwo-axis spectrometer^atEL³Saclay12(Figure^3).The principle ofthis apparatus^istodefine the divergence of the scattered beam by ^a system of antidivergent ^slits without changing the counter-sample distance ^{and to use mono-} chromaticincidentneutrons, thus avoiding wavelength^cor- rections.

(a) Monochromator. This is a monocrystalof pyrolitic graphite^at the exit ofa cold neutron _guide. It _produces a

neutron beam of_wavelength X ⁼ 4.62 ± 0.07

.

A second graphite monocrystal eliminates the higher order contami- nation. The angular divergence of the incidentbeam is 30 min.

(b) SampleHolder. Thisandthe oven are analogous to thosedescribedabovefortheDll _apparatus.

(c)

Antidivergent

^{slits. These}slits,placed betweenthe sample and the counter, ^are arranged to define the ^scat- tered beam falling ^{on the counter. The beam} defined by theseslits has a horizontaldivergence of³⁰min. Thescat- tered wave vector,q, varieson this_apparatusbetween 2X 10~2and2X 10_1Á-1.

Sample

Preparation

The preparationofthedeuterated polystyrenes^{was car-} riedout under high^vacuum byanionic polymerization. The absence of termination reactions during the polymeriza- tion,togetherwith ajudicious^choiceoftheinitiator ^andof the polymerization solvent, leads always to atactic poly-

mers exhibiting ^a known molecular weight ^and character- izedby ^{a narrow} distribution ofmolecular weights^{and ab-}

sence ofbranching.20

Table V

PSD PSH

Sample

no. Mw Afw/Mn

MjM.

1 21,000 1.05 1,05

2 57,000 1.17 1.14

3 90,000 1.20 1.17

4 112,000 1.35 1.23

5 160,000 1.13 1.13

6 325,000 1.80 1.40

7 500,000 1.14 1.18

8 1,100,000 1.17 1.18

Useofa nonpolar solvent andof_secondary

butyllithium

(BuLi) ^{as the}

initiator

are of_special interest for _{the poly-} merization of_styrene.21 Inthis case,the initiation andthe propagationstepsmay^beseparated by^an adequatechange of the temperature (at ^t below 5°, merely initiation takes place, while propagation ^becomes effective above 20°).

This procedure provides ^an easyway for removing the ^re- mainingprotonic impurities from ^{the monomer before} po- lymerization ^and thus offers the best available conditions for obtaining monodisperse polymers. Before ^each experi- ment, all_glass walls ofthe apparatus and the polymeriza- tion solvent^are carefully^washedwith a dilutedBuLi solu- tion.

Pure secondary BuLi is obtained by distillation under high^{vacuum at 80°} of the compounditselfafter having dis-

tilled off

the solvent.Dilution atthe desiredconcentration issubsequently achieved in the solvent^chosenforthe poly- merization.

Styrene, about ^99% deuterated, ^was provided by the CEA. The monomer isfirst distilled on Na wire on an ap- paratus fitted with ^a small Vigreux column, then it is dis- tilled twice under highvacuum, ^on Na wireandon molecu- lar sieves. In this _way, a reasonable _degree of

puriirity

(about ^{2 X 10~4}

/1.

^residual protonic impurities) ^{can be} reached without _wasting a large amount of the expensive monomer.

The solventchosenfor thepolymerizationin most exper- imentsisbenzene. Toluene, which^was first used, gives rise to transfer reactions with the polystyrylcarbanions22 (transfer ^{constant = 5 X 10-6 at} 50°) ^so thatthemolecular weightsofthe products obtained^are lowerthan thosepre- dicted from the ratio ofthe initial amounts of monomer

and

initiator,

^and the molecular weight distribution is en-

larged. ^As expected, this ^side reactionexhibits an increas- ing influence ^on molecular weight andpolydispersity^{as the}

monomer dilution (solvent concentration) ^andas the poly- styrene molecular weight (ratio styrene-initiator) ^become higher.

The weight-average molecular weightsofthe_samples of deuterated PS ^were determined by

light

scattering ^mea- surements inbenzeneor THF on a Fica apparatus. The ^re- fractive index increments, ^{measured on a} Brice-Phoenix differential refractometer, respectively^0.091 and0.174cm3 g_1, ^are noticeably lower than the corresponding incre- ments of hydrogenatedPS (0.106 and 0.198).

The polydispersityof the products^was characterized by gelpermeation chromatography, with ^use ofa WaterAsso- ciates apparatus combined with ^an automatic viscometer, THF being the solvent. The usual way of calculating the different_averagemolecular weights (Mn, Mw,Mz)fromthe GPC diagrams using the logM vs. elution _volumecalibra- tion does not fit satisfactorily, ^since calibration curves are

different for ordinary ^and perdeuterated polystyrene.

Polymerization

solvent _Mw Mw/M„

Toluene 20,000 1.50

Toluene 55,000 1.08

Toluene 83,000 1.07

Toluene 114,000 1.024

Benzene 172,000 1.02

Benzene 320,000 1.04

Benzene 530,000 1.10

Benzene 1,170,000 1.2

PSD 8

-

1,1,|Q6

*1-1.17

Mn

UMLV ^VLLL

Figure ^4.Gelpermeation chromatography. Elution^curve ofsam-

ple PSD^8.

Therefore we used the so-called “universal” calibration whichassumes that _[r¡]M,^measure oftheparticlesize, isin- dependent ofthe nature ofthepolymer.23 From the plots of _log

[ ]

^{vs. elution volume, we were able to calculate}

backthe_average molecular weights. Figure^{4 showsa}

typi-

calelution ^curve. The molecular weight^andpolydispersity

are collected in Table V. Molecular weights range from 21,000 to 1,100,000. The Mz value has been calculated, sincewe shallneed it for _correctingradii of gyration. This calculationhas not taken into account theaxial dispersion of the columns. It therefore _gives ^{an excess} value of the polydispersity. Inthe last column ofthis_table,we givethe characterization ofsome hydrogenated polystyrene (PSH) sampleswhichwere usedas matrices insome experiments.

The section of the beam in both apparatuses ^is of the order of 2 cm2 and determined the surface ofthe _sample.

Itsthickness0.8mm foraPSHmatrix and5mm for^aPSD matric has beencalculated in orderto have ^a transmission factor larger than ^0.5. Tobe sure that this relatively^small value does not introduce multiple scattering effects ^we have measured some samples with 0.4 mm thickness ^and obtained the^same result.

The solid samples ^are disk-shaped ^{(2 cm} in diameter) with concentrations of labeled chains rangingfrom0to 2 X 10-2_gcm-3.

These samples^were prepared by^two methods leading^to the same neutron scattering results. In the first method, samples^are obtained by castingdilutechloroform solutions ofPSD and PSHon a mercury surface.The evaporation of the solventis achievedunder ^vacuum during several days.

In the second method PSD-PSH mixtures are directly molded above Tg under ^vacuum, using ^a device designed by^J. P. Gabel and A. J. Kovacs. Thesemixtures are them- selvesobtained by^freezedrying^ofPSD-PSHsolutions.

ResultsandDiscussion

Typical^scattered intensities are shown in Figure ^5. The upper ^curve corresponds ^to the scattering by deuterated

Figure ^5. Experimental data obtained in ^the intermediate mo-

mentum rangefromthescatteringby the PSD ¹in theH matrix and by the H matrix (+). The increase on the left ofthe lower curve comes fromthe wingofthedirectbeam.

Figure^6.Intensity, in arbitraryunits, obtained^{as a}function of_q by differenceofthecurves ofFigure5.Thefulllineisacalculated

curve whichisexplained^{atthe end}of this_paper.

polystyrene^chains ofmolecular weight21,000,dispersed in

a matrix of undeuterated polystyrene ^chains. The lower

curve corresponds^to the scattering by undeuterated poly- styrene bulk material. Ideally ^{this curve should show no} dependence^on q, at least inthisrange. The increase atlow angle ^is due to the divergence ofthe incidentbeam, since

even withoutsample such ^an effectisobserved.Thediffer-

ence curve (Figure ^{6) is}the scattering law^{to be} interpret-

ed. The average momentum range covered goes from ² X 10-2 to 2 X 10-1 Á-1, which includes thus ^a small _angle scattering range q ^{< 1}_/Rg and an intermediate scattering range q > 1/Rg. The intensities were measured ^over the whole q range with the same apparatus (Figure ^3). For higher molecular weights, both ^momentum ranges per- formed couldnot be covered with one experimental setup andthustwotypes ofexperiments^were performed.

In thefirst, ^{we measured the} radii of gyration ^{from the} scatteredintensity^/(q⁾

=KcM{

^{1 -} (q2Rg2/3))for_samples

Figure^7.CoI~1 (E)^isplotted^vs. Co^+R2 whereR isafunction ofthe scatteringangle ⁼ 2tR/D. Theslopes ofthe curves give the uncorrectedradiusof gyrationand the slopeofthepoints (X) obtained foreach_{C d} byextrapolation^{atR = 0}givesthe value A2

ofthe secondvirialcoefficient. Thedataare obtained fromPSD4

initsPSHmatrix.

of molecular weight M as monodisperse ^as possible, with eightdifferentmolecular weights, in order^to test therela- tionship

R?

⁼

^KMue

⁽¹⁹⁾

whereK is a constant andca parameter

(it

^is recognized24 that ^{1 + e = 2v where v is a} characteristic critical _expo- nent) depending ^on the excluded volume. For this _experi- ment, in the small momentum range,^we used the appara- tus in Figure ^2. The resultswere compared with radii ob- tained fromthesame polystyrene^chainsandwiththesame

technique in^a goodsolvent (CS2) andina 6solvent (cyclo- hexaneHatT ⁼ 36°).

Thesecond seriesofmeasurements consistedinthe anal- ysis ofthe segmental configuration from the shape of the momentum _dependenceof the scatteredintensity_{for larger} valuesofq.Thefactthatthe_segmentsare connected inthe form ofachain yields^a characteristic scattering^law.Inthe absenceofexcludedeffect,Debye25 has shownthat

1(g) ⁼

KcM(2/x2)(x

^{- 1 +}

ex)

(20) with^{x =} _q2Rg2.

When x is large and if we write _Rg2 ⁼ _b2N/6, b being ^a characteristic length ^associated with the statistical ele- ment, N ⁼

M/m

^the ratio of the chain molecular weight

over thestatisticelement molecular weight,^one obtains

I{q)

⁼

KcM\2/q2b2{M/m) l/Rg

^{< q}

^< l/b

(21)

These measurements were performed in the intermedi- ate momentum range using theapparatus^shown in Figure

3.

(A) Small Momentum Range.Radii of

Gyration.

It is known that when there are concentration effects, ^one has to extrapolate the radii of gyration obtained ^at finite con-

centration of labeled polymersCq to^zero C. Thishas been doneusing the Zimm procedure for^a few_samples in cyclo- hexane and in the bulk. For small concentrations, Cq of deuterated chains, and for small values of _q the inverse

TableVI ñw,Á

PSD Mw cs2 Bulk ^C6H12(36°)

CeHi2 (35°)

1 21,000 ⁵⁰(E) 38(E) ⁴² (A)

2 57,000 84(E) ⁵⁹ (J) ⁷⁰ (B)

3 90,000 115 (G) 78(F) ⁸⁸ (B)

4 112,000 ⁸⁷(F)

5 160,000 ^{168 (G)} 107 (G, K) ¹¹⁷ (B) ¹⁰⁸

6 325,000 204 (G) 143(C,F,H) ¹⁵⁰ (B,C)

7 500,000 ²¹³

( , , )

^{191 (C)}

8 1,100,000 ⁵⁶⁸ (I) 297(1,K) 293 (D) 303

Technique ^{NS“ NS“ NSa LS6}

NS⁼ neutron scattering.6 LS⁼ lightscattering.

scattering intensity ^{can be} represented by the Zimm for- mula

CdKI-H6)

⁼

¿(1

^{+ q2Re2/3) + 2A2CD (22)}

andthe valueof_Rg2isobtained byextrapolationto Cq⁼ 0.

Figure 7, for instance, shows the Zimm plot ^{which corre-} sponds to PSD 3. Similar _diagramshave beenobtained on

samples 1, 4, 6, and 7. Within the experimental ^{errors no} concentration effect can be detected. This result allows a

simplification of the experimental procedure. In effect all theother_sampleswere measuredat only^one concentration Co ⁼ 10-2 _g cm-3, in the cyclohexane H solutions ^{at 36°}

and in the bulk.Only in CS2have we extrapolated to ^zero concentrationaftermeasurements atfourconcentrations.

It iswellknownthat scattering experimentsdo not give^a weight average for the radii ofgyration. More precisely if

we assume between fig2 andM the relation of_{the type} in

eq 19, the experimental radii of gyrationcorrespond ^{to an} averagemolecular weight given by therelation

This_average isa “z” _{averageonly if} c⁼ 0. Sincewe can ex-

pect ⁽ to be small ^{as a} first approximation, ^which will be

justified later, ^we replace

(M)

byMz. Unfortunately, ^the precision ofMz ^ismuch lowerthan that ofMw. We there- foreprefer^to plot^{our resultsas} functionsof_{Mw,correcting} the radii of gyration by thefactor (Mz/Mw)1/2. This factor being in most ofthe cases rather _small, both methods are

equivalent. The ^valuesobtained, which ^{we have}called_Rw,

are reported in Table VI. The capital letters in brackets correspond^tothe experimental setup (seeTableIV).

In_Figure8we haveplottedlog^{R as a}function of_logMw for the three cases: deuterated polystyrene in ^a matrix of hydrogenated polystyrene, polystyrene in cyclohexane ^at 36°, and polystyrene in carbon disulfideat room tempera- ture.

The first striking result^isthat inbulkthe dimensionsof the molecules^are nearly the^{same as} in cyclohexane(CgH^

at 36°) in the entire range of molecular weights. This strongly supports the hypothesis that the sizes of the chainsare identicalin both situations.Sincecyclohexane^is at 36° a solventfor polystyrene, thecoefficientin_{eq 19 is}

zero and_Rg2should_obeytherelation

R}

^{= KM (24)}

and actually in Figure ^{8 we have} drawn a straight ^{line of}

Figure^8.Log^{Rw is}plotted^vs. logMw. The experimental data^are obtained indifferentenvironments: (X) in^agoodsolvent_CS2,(+) ina6solvent,

( )

in the bulk. The_slopesof0.6(inCS2) andof0.5 are obtained by^abestfitmethod.

Figure^9. (Rw2/M„)1/2^isplotted^{as a}function oflogMw.The data

are (X) for^PSD in_CS2,(+) in solvent, and

( )

^{inthebulk. The}

horizontal line, which^isthemean for the data obtained in sol- vent andinthebulk,^isconsistentwith aGaussianconfiguration.

slope^0.5which_goesthroughthe experimental points. Fig-

ure 9shows (RW2/MW)1/2^asafunction of_{logMw. The}hori- zontalstraight^linegives the valueof_(K)1/2 ⁼ 2.75 X 10-9 cm.

We note that the best fit for the cyclohexane poifits would not give^a slope exactly equal^to 0.5, butthis is not surprising. For ordinary polystyrene it is well known that incyclohexane the point^isofthe orderof_35°; thiscould beslightly different fordeuterated polystyrene.It has been shown by Strazielle, ^et al.,26thatthereis asmalldifference in temperaturefordeuterated polystyrene (PSD) inordi- nary cyclohexane. Moreprecisely, Strazielle^hasfoundthat for PSD in the point^is30°, forPSH in CeDi2 it is 40.2°, and for PSD in _C6Di2 it isof the order of 36°. This small difference in temperatures (30° instead of _36°) should ^be taken into account for _{very precise} measure-

ments. From viscosity data ^{we can} evaluate its effect as

being ofthe order of 4%on the valueof _Rg.This couldex-

plain why in cyclohexane theRgvaluesdo not vary exactly

as in thebulk.

It should ^be noted that this comparison between ^values obtained in cyclohexane and in bulkisveryprecise,^even if there issome systematic ^error in the absolute values, since measurements have beenmadein bothcases withthesame

apparatus using the ^same corrections. Nevertheless, it ^was also important ^{to check} if the absolute values of radii of

Figure ^10.q2I(q) in arbitrary units isplotted ^us. q, obtained for different_samplesindifferentenvironments: (A)forPSD1in PSH matrix, (B) for^aPSH_(Mw⁼ _114,000)inamatrix ofPSD_4,(C)for PSD6in its PSH matrix, (D)foraPSH_(Mw⁼ 3.8 X 106)in sol- vent (CeDi2 40°), and (E) forPSD 10in its PSH matrix. Thefull linesare calculatedcurves obtainedfrom the_Debyefunction.

gyration^{have beenmeasured}correctly. Forthis_purposewe

have measured the radius of_gyration for two samples by

light

scattering. ^Since the principles ofthe measurements

are exactly the ^same and sincethe^same type ofmolecular weightaverage ^isobtained, the agreement should ^begood.

For_samples 5and8 incyclohexane (30°) ^{we measured}the values of 108 and 303 Á, respectively, which ^are in _very good agreementwith the neutron results. It iswell known that carbon disulfide is a good solvent for polystyrene, therefore thecoefficientt from_{eq 19 is}positive and should reach its asymptotic value for _large molecular weights ^e 0.2.Thebeststraightline (Figure8)gives

R?

^{= KMU2}

which is very satisfactory. This result demonstrates the possibilitiesoftheneutron scattering techniqueforabetter experimental verification of viscosity theories, ^since it is possible to ^measure simultaneously the

limiting

viscosity indexandthe dimensionsofthe molecules.

Effect

of Temperature. It ^{is wellknown}thatbelowthe glass transition temperature polymeric systems ^are not in equilibrium. ^One objection ^{to our} resultscould be that in these experiments ^{we are} measuring frozen conformations unrelated to equilibrium properties ^and that the _good agreement^betweenresultsobtainedin_CeHi2andinbulkis

fortuitous. Wewouldliketo recallthat^we usedtwoprepar- ative techniques. Inthesecond one wherethe disksare ob- tained by molding,allthesamples have been annealedfora

fewhours, above _Tgat 120°. No differences have been de- tected between the results obtained for the two types of sample.

To prove^more directly that the dimensionswe are mea-

suring correspond ^to equilibrium conformation we have made measurements on one ofour samples, 6,at 120°,hav- ing waited³⁶hrto besure equilibriumhad been reached.

At room temperature ^we obtained 143 Á for _Rg and at hightemperature^134Á.This small difference,8%,couldbe due to experimental^error. It couldalsobeexplained by the thermal effect ^on the statistical unit ^or unperturbed di- mensions. If it is confirmed that in the bulk state volume effects can be neglected, this kind of experiment should

Figure^11.q2I(q)^isplotted, inarbitraryunits,^us. q. The_samples

are PSH (Mw ^{= 1.1 X} 106)inaPSD10matrix, forthe concentra- tions(2.00,1.52,1.01,and0.512 X10-2gcm-3). Theanomalous de-

crease at thebeginning of^{thecurves isnot an} effectof multiple scatteringsincethe_sampleC p ^- 1.52 X 10-2withathicknessof2 mm (cross points)gives the^same resultas thesame sample of4 mm thickness.

allow study of thermal variation of unperturbed dimen- sionsover awide_rangeoftemperature.

Interaction between Chains. For molecular weights of samples1,4, 6,and7,the intensities corresponding^to four finite concentrations were measured, showing that the coefficient A2 in eq 22 is always lower than5 X 10-5 cm3 g-2. This raises the problem ofthe absence ofany visible coil interaction. Since there is no chemical difference be- tween the deuteratedandtheundeuterated_chains,thepair correlation g(r) ^{between two deuterated} segments sepa- rated by^adistancer shouldbewrittenas

g(r)

⁼

uD[p(r)

^{+ (1 -}

p(r))vO]

(25) where vd is thefraction ofdeuterated ^us. total number of segments and

p(r)

^thepair correlation _along a chain. The only hypothesis inthis formulationistheuniform distribu- tion of labeled _segments inthe matrix. The scattering in- tensity^isproportional^{to the}Fouriertransformof_g(r) and thisleadsto

(

⁼

MK2S(q)^(

¹

-

(Cd/p))

⁽²⁶⁾ wherep isthe densityofpolystyrene segments, Cd/p ⁼ m, and S(q) ^is the scattering law ^associated with a single chain. Formula 26hasthe expected symmetry in substitu- tionofCd/pfor₍₁ ^— Cd/p)). There^isno scatteredintensity inthelimits Cd ^{= 0}and Cd ⁼ p. Inthelimit _q ⁼ _0, thein-

verse scatteredintensity^isthen

C^rHq)

⁼

^(l/M)

⁺

(Cd/M)

(27) Comparingwith_{eq 22,}we find^asecondvirialcoefficient

A2 ⁼

1/pM

(28)

ThisvalueofA2 isalwayssmallerthan5X 10-5cm3g-2for the molecular masses used in our experiments ^andthere- fore_{eq 28 is} not incompatible withthe small experimental valuesofA2.

In thisdiscussionwe haveshownthattheradiusof_gyra- tion in thebulk isequal ^to the radiusof gyrationin cyclo- hexane for the same molecular weight ^and that _A2 ⁼ 0.

This strongly supports the hypothesis of the Gaussian characterofthe polystyrene^chains in the bulk. Neverthe-