Density ¯uctuations in amorphous systems: SAXS and PALSresults

L. David a,*, G. Vigier a, S. Etienne b, A. Faivre c, C.L. Soles c, A.F. Yee c

a GEMPPM, INSA de Lyon, UMR CNRS 5510, 20 av. A. Einstein, 69621 Villeurbanne cedex, Franceb LPM, Ecole des Mines de Nancy, UMR CNRS 7556, Parc de Saurupt, 54041 Nancy cedex, France

c Department of Material Science and Engineering, The University of Michigan, Ann Arbor, MI 48109 2136, USA

Abstract

The electronic density ¯uctuations in polymethyl methacrylate (PMMA) were studied by synchrotron radiation small

angle X-ray scattering (SAXS), and compared with the hole volumes as revealed by positron annihilation lifetime spec-

troscopy (PALS). The results of both methods are shown to be compatible with a structure of the liquid and the glass

with local heterogeneities of radius of the order of 0.3 nm. These microstructural investigations thus support the exis-

tence of density ¯uctuations in amorphous systems on a nanometer scale. Ó 1998 Elsevier Science B.V. All rights re-

served.

1. Introduction

The concept of heterogeneity in liquids andglasses is invoked in several theories of glass tran-sition [1,2] often through the relation between theglass transition relaxation time and the state of dis-order of amorphous systems. Although manycomplementary spectroscopic techniques can leadto a detailed description of the dynamics in glassesand liquids at various length scales, there is nohope to obtain as complete a picture of the struc-ture from current microstructural characterizationtechniques.

The aim of this work is to present and analyse re-sults provided by small angle X-ray scattering(SAXS) and positron annihilation lifetime spec-troscopy (PALS), to contribute to the under-standing the structure on a nanometer scale ofamorphous polymers in the liquid and glassy states.

2. Experimental techniques

Small angle X-ray scattering (SAXS) measure-ments were performed at the ESRF (Grenoble,France), on D2AM beamline. Synchrotron radia-tion was required here to obtain spectra in thescattering vector �q � 4p sin�h�=k� range from5� 10ÿ3 to 1 �A

ÿ1in the short time exposure of

60 s, thus providing time resolved accuracy inpoint collimation conditions during a heating runat 3 K/min. A linear position sensitive methane±xenon gas detector was positioned at about 40cm of the sample.

The positron annihilation lifetime spectroscopy(PALS) spectra were also obtained as a function oftemperature. For each spectrum, a 35lCi22NaClsource was sandwiched between two identicalsamples. The data were collected by means of amultichannel analyser supported (MicroVaxscomputer (DEC, 3100)), and the lifetime and in-tensities were calculated by means of the PFPOS-FIT software.

Journal of Non-Crystalline Solids 235±237 (1998) 383±387

* Corresponding author. Fax: +33-4 72 43 85 28.

0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 9 8 - 5

The polymethyl methacrylate (PMMA) samplesused for SAXS and PALS experiments were pur-chased (Goodfellow (Clinical Quality, ref.ME303011)). The calorimetric glass transition is388 K assessed from the onset point determinedat 10 K/min heating rate.

3. Results

3.1. SAXS measurements

The lower q range of the spectra obtained forPMMA were ®tted to the usual empirical law

I�q� � I0 exp�bq2�: �1�This procedure yields the evolution of I0 and b as afunction of temperature, as displayed by Fig. 1.The increase of I0 with temperature through theglass transition is well known [3], but the evolutionof b is less documented [4]. b is positive in thewhole temperature range investigated, and de-creases with temperature. The order of magnitudeof b is a few �A

ÿ2. We observed a similar change for

a wide variety of amorphous systems, either poly-meric (PS, PC, PES) or molecular (sugars likesorbitol or maltitol [5]). In the case of PMMA, bis close to 3.5 �A

ÿ2in the vitreous state and decreas-

es above the glass transition.

3.2. PALS measurements

The evolution with temperature of the ortho-Positronium (o-Ps) lifetime, s3, and the corre-sponding relative annihilation intensity, I3, isshown by Fig. 2. s3 is related to the size of theempty regions, whereas I3 is usually assumed tobe proportional to the number density of holes.Both quantities increase with temperature, as usu-ally observed for other polymers such as PS andPC. In the case of PMMA, s3 is close to 2.2 nsat Tg, and I3 exhibits a plateau above Tg. These re-sults show that the number and the size of holes in-crease with temperature up to Tg.

4. Discussion

4.1. Determination of ¯uctuation sizes by SAXS

As the value of parameter b is positive, it is notpossible to extract the size of the scattering entitiesby considering a collection of diluted particles,scattering at low angles according to the Guinier'slaw. Interference e�ect thus may play a dominantrole. Keeping the hypothesis of a particle-liketreatment of the electronic density ¯uctuations,the scattered intensity is thus given (in electronunits) by

I�q� � NF 2�q�S�q�; �2�Fig. 1. Evolution of the extrapolated intensity at zero angle I0

and slope b of the spectra in log �I�q�� ÿ q2 plots, as a function

of temperature during a heating run at 3 K/min.

Fig. 2. Evolution with temperature of the ortho-Positronium

(o-Ps) lifetime s3 and the relative annihilation intensity I3 as a

function of temperature.

384 L. David et al. / Journal of Non-Crystalline Solids 235±237 (1998) 383±387

where N is the number of ¯uctuations in the scat-tering volume, V, F 2�q� is the scattered intensity byone particle, and S(q) the particulate structure fac-tor. If identical homogeneous spherical particlesare assumed to represent the ¯uctuations, thenF 2�q� is given by

F 2�q� � v20S Dq2 3

sin�qRS� ÿ qRS cos�qRS�q3R3

S

� �2

� v20S Dq2 U2�qRS�; �3�

where Dq2 is the contrast factor and v0S and RS are,respectively, the volume and the radius of eachparticle.

The structure factor is written as

S�q� � �1� 8v0S=v1SU2�nSq��ÿ1

: �4�nS is the characteristic distance between the parti-cle centres, and v1S � �nS�3 is the average volumecontaining one scattering particle. This expressionis known to describe a variety of non-crystallineparticulate systems, such as colloids and organic/inorganic hybrid materials [6]. Moreover, in thecase of ¯uid of hard spheres, the expressionEq. (4) is exact with nS � 2RS [7].

A lower q expansion of I(q) given by Eqs. (2)±(4)is thus possible, leading to expression Eq. (1) with

I0 � a16p2

9

R6S

n3S I � 32p

3:

R3S

n3S

� � with a

� 16p2

9CSV Dq2; �5�

b � R2S

5ÿ 1� 16p

3 nS

RS� 32p RS

nS

� �2

0B@1CA: �6�

The ®rst negative term of Eq. (6) is the usualGuinier's law contribution, i.e. is due to the scat-tering of diluted particles, but there is also a posi-tive term arising from the structure factor. Otherminor contributions are neglected [3], such asatomic scattering factors. In principle, the mea-surement of absolute intensity spectra for whichCS � 1, together with the knowledge of V andDq2, lead to the determination of RS and nS. Inpractice, even if absolute intensity measurementsare possible with polyethylene (Lupolen) calibrat-

ed samples, V and Dq2 are often estimations. Onthe contrary, b is not sensitive to absolute measure-ments. b is positive when y � nS=RS is <18. More-over, PALS measurements of Fig. 1 indicate that Rshould increase and n should decrease with increas-ing temperature, implying that the ratio, y, shoulddecrease with temperature while b decreases, i.e.ob=oy > 0. An examination of Eq. (6) shows thatthis change is possible only when y is small, i.e.nS=RS between 2 and 4. As a consequence usingb � 3:5 �A

ÿ2, an estimation of RS can be found to

lie between 0.3 and 0.5 nm. The scattering entitiesare thus local ¯uctuations of electronic densitywith radius <1 nm. (A similar conclusion is alsoreached by directly assuming nS=RS < 10, whichis considered to be always the case.)

4.2. Determination of ¯uctuation sizes by PALS

The procedure yielding the size and numberdensity of hole volumes [8] will not be presentedin detail here. We shall only recall that the radiiof spherical holes is related to s3 by

1=s3 � kspinG� ko-PS�1ÿ G�;where G � 1ÿ RP=�RP � DR� � 1=2p sin�2pRP=�RP � DR��: Quantities kspin, ko-Ps, DR have stan-dard values [8]. Moreover, the total hole volumefraction is related to I3 by

FP � v0P=vIP � CPv0PI3;

where CP is a constant determined by the use of di-latometry measurements [8].

Fig. 3. Changes with temperature in the radius of hole volumes

RP, and volume fraction of holes FP as deduced from PALS

measurements according to Ref. [8].

L. David et al. / Journal of Non-Crystalline Solids 235±237 (1998) 383±387 385

The evolution of FP and RP with temperature isshown by Fig. 3. The average radius of the holes,RP, is of the order of 0.25 nm, and the mean dis-tance between the holes, nP � �FP=v0P�1=3

is about3±4RP. These numbers are in good agreement withthose deduced by SAXS in the preceding section.This agreement leads to the conclusion that thehole volumes are a dominant contribution to thescattering.

4.3. Further use of SAXS data

The RP and nP deduced by PALS at a given tem-perature (for example 213 K) can be used for thenormalization of scattered intensity by means ofEqs. (5) and (6). Indeed, a and k � RS=RP can becalculated. RS and FS � v0S=v1S at 213 K are thusidenti®ed respectively, to the kRP and FP at thesame temperature. The corresponding solutionfound for k is close to 1.2, indicating again thatthe sizes of electronic density ¯uctuations is largerthan the sizes of the holes. Lastly, Eqs. (5) and (6)can then be combined yielding RS and FS at anytemperature from the measurements presented inFig. 1 and a. The results are displayed in Fig. 4.They are again in good agreement with Fig. 3,showing evidence that both techniques are sensi-tive to similar entities. The size of the scattering¯uctuations SAXS is nearly constant in the tem-perature range investigated, whereas the size ofthe holes PALS increase with temperature. More-

over, in both cases, the volume fraction of the elec-tronic density ¯uctuations or holes increase withtemperature in a similar way.

5. Conclusion

The density ¯uctuations that are present inamorphous polymers, and more generally in or-ganic glasses are small volumes of about 100 �A

3,

as deduced by PALS and SAXS measurements.These measurements are in good agreement withearlier estimations on the basis of SAXS data [9]and measurements of hole volumes by ¯uorescentand photochromic probes techniques [10]. Thiswork does not show evidence of larger scale heter-ogeneities as probed by other techniques such aslow frequency Raman scattering [11]. In spite ofthe approximations made in both data treatmentsand the di�erences in the nature of the ¯uctuationsrevealed by SAXS and PALS, the evolution withtemperature of (i) the characteristic radius of theholes and of electronic density ¯uctuations and(ii) the volume fraction or the characteristic dis-tance between ¯uctuations/holes are similar. Nev-ertheless, there are fundamental di�erencesbetween the two techniques, in that the smallestvolumes (R < 2 �A) are not detected by PALS,and SAXS is sensitive to positive and negative den-sity ¯uctuations [8]. However, the hole volumes ev-idenced by PALS are probably the most apparentscattering entities, as evidenced by the similaritiesdiscussed above. The results presented here arethus in agreement with a description of amorphoussystems relying on small scale nano¯uctuations ofdensity [2,12].

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