diffusing-wave-spectroscopy investigation of latex particle motion in polymer gels

PHYSICAL REVIEW E MARCH 2000VOLUME 61, NUMBER 3

Diffusing-wave-spectroscopy investigation of latex particle motion in polymer gels

G. Nisato,* P. Hebraud, J.-P. Munch, and S. J. CandauLaboratoire de Dynamique de Fluides Complexes, 3 rue de l’Universite´, 67084 Strasbourg, France

~Received 1 February 1999; revised manuscript received 5 November 1999!

Dynamic light scattering studies of particle motion in cross-linked chemical gels are limited by the fact thatwhen the cross-linking ratio is increased, the scattered intensity becomes quenched, and time-dependent con-centration fluctuations contribute less and less to the overall scattering signal. The concentration fluctuationsthen occur at very small length scales, which can become inaccessible to classical dynamic light scatteringtechniques. Diffusing wave spectroscopy can help overcome these experimental difficulties. We extend thistechnique to the case of nonergodic systems, and apply it to the detection of the motion of probe particlestrapped inside chemically cross-linked gels. We measure the mean square displacement of the nanometricprobes, leading to the measurement of local mechanical properties of the gel matrix.

PACS number~s!: 83.85.Ei, 83.10.Dd, 82.70.Gg

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I. INTRODUCTION

Microrheological techniques have been applied tostudy of soft-matter systems since the late 1920s, and hrecently received an increasing amount of interest, both frexperimental and theoretical points of view@1–6#. The mainadvantage of these techniques resides in the fact thatallow anin situ determination of the rheological propertiesa wide range of materials extending the dynamic rangeclassical rheometers as well as the type of systems, syntor biological, that can be studied. Recently, several wofocused on ‘‘passive’’ microrheological techniques@4,5,7#.In this kind of approach, thermal fluctuations drive the mtion of probe particles embedded in the medium being stied. By measuring either the mean square displacementhe power spectrum of the position fluctuations of the proparticles, it is possible to calculate the mechanical respoof the medium. A classic example is given by the Brownimotion of latex spheres suspended in a Newtonian fluid:Stokes-Einstein relationship relates the mean squareplacement of the particles to the viscosity of the mediuwhich can be thus measured once the particle size is kno

When the medium surrounding the probe particles is etic ~i.e., has a nonzero storage modulus at zero frequen!,the elementary theory of elasticity states that the particlesgeometrically confined. The inherent nonergodicity of thesystems has to be addressed when light scattering technare used to measure the motion of probe particles. Inpaper we address this point by extending the formalismwas developed for interpreting dynamic light scatteringnonergodic media to the case of diffusing wave spectrcopy, and we also report the results of a study on a mononergodic system.

The study of the dynamics of nonergodic systems is mdifficult by the fact that they explore only a fraction of atheir possible configurations. More specifically, the expementalist who wants to investigate the domain of the phspace that is actually explored by the system and its ass

*Present address: Philips Research, Prof. Holstlaan 4~WB72!,NL-5656 AA Eindhoven, The Netherlands.

PRE 611063-651X/2000/61~3!/2879~9!/$15.00

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ated dynamics encounters two main difficulties. First of aensemble averages over all the configurations have to beformed explicitly by the observer. Second, the length scafreely explored by the system are limited, and therefore inecessary to perform observations at short time and lenscales.

Polymer gels are a very good example of nonergodic stems. In these materials, the concentration fluctuationsgeometrically and topologically constrained by the preseof cross-links that act as a quenched disorder. The dynaproperties of polymeric gels have been investigated usdynamic light scattering~DLS!, which probes the time-dependent density fluctuations occurring in the scattervolume under observation@8–10#. The scattered intensity isfound to be the sum of a random thermal component anquenched term that depends on the location of the scattevolume inside the gel@11#. The intensity associated with thfrozen-in component depends in a complex manner onconditions of preparation of the gel. More specifically, itrelated to the polymer concentration and to the cross-linkratio, defined as the molar ratio of cross-linking units to ttotal number of monomers.

Upon increasing the cross-linking density, the thermmotion of the scatterers becomes more and more hindeand one needs to probe the system at smaller and smlength scales to detect any motion or relaxation phenomeEventually, the scale of observation is no longer accessto DLS, and one must turn to other techniques of investition. Diffusing wave spectroscopy~DWS!, an extension ofthe light scattering technique suited for the study of highscattering systems, allows one to probe displacements oscatterers of the order of a few angstroms. This is due tofact that the phase shift between an entering and an exphoton results from a large numberN of scattering eventsThe motion of each of the scatterers over a distance oforder of l/N is enough to completely decorrelate the sctered signal@12#. The motion of latex spheres suspendedwater and polymer solutions at length scales as small asangstroms has been investigated using this technique@13#.

In the present study we consider covalently cross-linkgels containing nanometric polystyrene spheres with diaeter much larger than the mesh sizej ~; 5 nm! of the poly-

2879 ©2000 The American Physical Society

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2880 PRE 61NISATO, HEBRAUD, MUNCH, AND CANDAU

mer network. The distance over which the particles cmove in a polymer matrix is controlled by the competitiobetween the thermal energy and the elasticity of the maAs the cross-link density is increased, the elasticity ofpolymer network increases, leading to increasingly smaspatial excursions of the particles, that eventually cannodetected by DLS. The amplitude of the mean squareplacement of each bead is limited and reaches a finite vaDr 2(`)56d2, at long times. Thus, when probing the motioof the beads at length scales larger thand, one can observethe effect of nonergodicity of the system: the result ofgiven light scattering experiment depends on the particscattering volume observed or, equivalently, on the particpoint of the speckle pattern being observed. One thereneeds to perform an ensemble average in order to obthermodynamically relevant quantities. The simplest amost efficient, but also must tedious method, consists offorming the ensemble average ‘‘manually,’’ by measurithe desired quantities for a great number of different confirations and then properly averaging them. However, itpossible to find relationships between measured time aaged and ensemble averages of interest. Pusey andMegen derived such relationships in the case of DLS@14#. Inthe first part of this paper we show that the same approcan be extended to DWS. We then experimentally invegate the dynamics of latex beads embedded in chemiccross-linked polyacrylic acid gels. By varying the cross-lidensity, one can tune the elastic modulusG in a rather largerange (;10 Pa,G<103 Pa). We took advantage of this posibility to determine the ranges ofG for which either DLS orDWS can be used, thus showing the complementary chater of these two techniques to probe the motion of the pticles trapped in polymer gels over a large gamut of lenscales.

II. DWS IN NONERGODIC SYSTEMS

The following presentation requires a certain familiarwith the DWS formalism presented by Weitz and PineRef. @12# as well as the work of Pusey and Van Megen@14#.We took advantage of the following analogies: one optipath in DWS plays the role of a single scatterer in DLS, aone distribution of paths in DWS plays the role of one sctering volume in DLS.

Let us consider a plane wave traveling through a medturbid enough for the phase of an entering photon to bedomized when it leaves the sample. In other words, we sthat each photon follows a random walk inside the medi@15#. The electric fieldE(r0 ,t) at the output of the samplcan be written

E~r0 ,t !5 (pPP

E0eiFp~ t !5 (pPP

Ep, ~1!

whereP is the ensemble of optical paths that begin atentering surface of the sample and ends at a pointr0 on thedetector,E0 is the incident electric field,Fp(t) is the phasedifference due to diffusion along pathp inside the sampleand Ep is the electric field associated with pathp. The ex-pression of the electric field is similar to the one encountein DLS,

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E~q,t !5 (r j PVdiff

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wherer j are the positions of the scatterers belonging toobserved diffusing volumeVdiff , andq is the transfer vector.However, Eqs.~1! and~2! differ in two aspects. In DWS, theexpression of the phase difference is a sum over a lanumber of scattering events, whereas only one scatteevent occurs in DLS. Also, in DLS one does not have anapriori information about the spatial correlation of the indvidual scatterers (r j ), whereas in DWS one may assume thtwo optical paths are not correlated@12#. As we will seefurther, the last property simplifies the expression of thenamic structure factors. It also leads to a linear relationsbetween the correlation function of the electric field assoated with pathp and the correlation function of the totaelectric field, which simplifies the computation of the latt@12#.

We now focus our attention on the expression of thetensity scattered at one pointr0 of the output. As all the pathsare independent we have

I ~ t !5 (pPP8

Ep~ t !•Ep* ~ t !. ~3!

If the system is ergodic, thenFp(t) is a random variable aswell asEp(t). The total electric fieldE(t) is then a Gaussianvariable, according to the central limit theorem. Therefothe ensemble-averaged correlation function of the intenmay be factorized according to@16#:

^Î ~ t !I ~ t1t!& t&«5Î ~ t !&«1@F~ t !#2, ~4!

where

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(p8

E02^êxp@ iFp~ t !2 iFp8~ t1t!#& t&« . ~5!

Since no correlation exists between any two given paths,ensemble average in Eq.~5! is redundant, as it is performeby the summation over the paths; hence

F~t!5(p

E02êxp@ iFp~ t !2 iFp~ t1t!#& t5Î &« f ~t!.

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By definition @14#, f (t) is the normalized dynamic structurfactor function. This relationship is of crucial importanceit links an ensemble-averaged function to a time-averaobservation at one given point, which is position dependeIt lies in the fact that all the optical paths have been assumto be noncorrelated.

Let us now consider a nonergodic medium. We assuthat each particle can move around a fixed positionRj , sothat its position may be written

r j~ t !5Rj1dj~ t ! ~7!

where^dj (t)& t50, so that the phase differenceFp(t) alongpathp may be decomposed in the same way:

Fp~ t !5CcP1F f

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PRE 61 2881DIFFUSING-WAVE-SPECTROSCOPY INVESTIGATION . . .

The first term in Eq.~8! is time independent, and the meavalue of the second one is zero. We are thus allowed to w

Ep~ t !5EcP1Ef

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where, if we putwp5êxp@iFfp(t)#&t , the fixed and the fluc-

tuating components of the electric field are

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EfP~ t !5E0 exp~ iCc

p!@exp~ if fp!2wp#. ~11!

The total electric field is the sum of a constant field andGaussian field:

E~ t !5 (pPP

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pPPEf

P~ t !. ~12!

Equation ~12! describes a heterodyne DLS experiment,which a reference field is superimposed to the field of sctered radiation@17#. In our case, the ‘‘reference’’ and thoscillating sources are a consequence of the nonergodicithe physical system, leading to a self-heterodyne situa@8#.

The experimentally accessible quantity is the timaveraged autocorrelation function of the scattered intenswhich can be expressed as@17#

Î ~ t !I ~ t1t!& t5Î f~ t !I f~ t1t!& t

12I cÊf~ t !•Ef* ~ t1t!& t12I cÎ f~ t !& t1I c2,

~13!

whereI f and I c are the fluctuating~time-dependent! and thefrozen-in components of the scattered intensity, respectivThe first two terms of Eq.~13! may be evaluated using thGaussian character ofEf(t), leading to:

Êf~ t !•Ef* ~ t1t!& t5Î &«@ f ~t!2 f ~`!#, ~14!

Î f~ t !I f~ t1t!& t5Î &«2@ I 2 f ~`!#21@ f ~t!2 f ~`!#2.

~15!

Most commercially available autocorrelators measurenormalized time-averaged autocorrelation function ofscattered intensity:gt

2(t)5Î (t)I (t1t)& t /Î (t)& t2, which is

linked to the structure factorF(t) through

gt2~t!511Y2@ f ~t!22 f ~`!2#12Y~12Y!@ f ~t!2 f ~`!#,

~16!

whereY5Î &« /Î & t . We remind the reader thatÎ & t dependson the on the spot of the speckle pattern being observed.relationship between time-averaged autocorrelation funcand dynamic structure factor expressed in Eq.~16! was es-tablished by Pusey and Van Megen in the case of DLS,discussed in detail in Ref.@14#.

If we assume now that the mean temporal value ofintensity does not depend on the observation point, we ob

~gt2~0!21!Î ~ t !& t

252I f212Î ~ t !& tI f . ~17!

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The left-hand side is thus a linear function ofÎ (t)& t , and itsslope is 2I f . This equation allows a direct way to checexperimentally the ‘‘self-heterodyning’’ of the scattering.also provides an accurate method to measureI f and to showit is not dependent on the sample position.

III. RESULTS AND DISCUSSION

A. Sample preparation

We studied the motion of latex particles embedded inmatrices chemically cross-linked to different degrees. Tprobe particles were latex particles~Poly Science nano-spheres!, their diameters were measured by DLS in wasuspensions at a volume fractionf52.531022. We used85-nm particles for the samples studied by DLS, and 107-particles for the samples studied by DWS. The spherescupied volume fractionsf51024 andf52.531022 in thesamples studied by DLS and DWS, respectively. Thematrices were polyacrylic acid~PAA! gels. The total poly-mer concentrationCp50.05 g/cm3 was kept constant in althe samples. The cross-linking ratioRc , defined as the molaratio of cross-linking molecules to the total molar monomconcentration, varied from 1.631023 to 131022 ~see TableI!. The samples were prepared by radical copolymerizaof acrylic acid and methylene bis-acrylamide monomersaqueous solutions containing the appropriate volume fracof polystyrene nanospheres. The radical polymerization retion, performed at 70 °C, was initiated by the additionaddition of ammonium persulfate immediately after the slutions were degassed. The samples were prepared directhe light scattering cells. The sample preparation was pformed along the lines of the experimental protocol detaiin previous publications, where further details can be fou@18#.

The polymer network thus formed is weakly anionic, rducing the risk of physisorption on the surface of the laparticles, since the latter are lightly sulfonated. For the pomer concentrations under consideration, it is possible totimate an average ‘‘mesh size’’j of the polymer network ofthe order of 5 nm, much smaller than the diameter ofprobe particles@19#. The polymer matrix can therefore, infirst approximation, be considered as a continuous mediumthe length scale of the nanospheres.

TABLE I. Fitting parameters and estimated local rheologicquantities @see Eqs.~23! and ~24!# corresponding to differentcrosslinking ratiosRc . DLS and DWS data were obtained witparticles having diameters of 85 and 107 nm, respectively.

Rc

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DLS 1.631023 390 144 0.52 1.9 10231023 250 89.2 0.47 1.8 16531023 26 14.4 0.31 2.9 150

DWStransmission

531023 16 8.89 <0.3 2.3 250

DWSbackscattering

131022 2.6 0.79 <0.3 1.2 1500

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B. Experimental setup

1. DLS

The coherent source was a Spectra Physics 2020 laoperated at 0.4 W withl5488 nm. The correlation functionwere calculated by an ALV 5000 logarithmic autocorrela~in the ‘‘Single’’ mode!. The coherence factor of the expermental setup was measured with a reference latex suspeand corresponded tob50.9760.02. All the experimentswere performed in a controlled environment at 22 °C60.5.The scattering cells were vertically translated with a Pcontrolled step motor~Microcontrole!, allowing reproduciblepositioning with a resolution of 1mm. The total excursionwas 7 mm. For each sample and scattering angle, we msured 1000 different autocorrelation functions of the sctered intensity~10-s acquisition time for each one of them! inorder to calculate the ensemble-averaged dynamic strucfactor @9,10,14,20#.

2. DWS

The coherent source was a Spectra Physics 2020 laoperated atl5488 nm. The laser beam was expandedapproximately 1 cm at the sample. The diffused light wcollected by two monomode optic fibers~ALV ! connected toALV photon-counting devices. One of the fibers was placin the axis of the beam~‘‘transmitting’’ geometry!, and thesecond fiber was placed in the backscattering cone~‘‘back-scattering’’ geometry!.

The characteristic length scales probed by DWS in‘‘backscattering’’ and ‘‘transmitted’’ geometries are defineby l B

25@1/k02(z0 / l * 12/3)2# and l T

25( l * /k0L)2, respec-tively, wherek0

252p/l, L50.5 mm is the thickness of thsample,l * 5140mm is the scattering mean free path, andz0is the penetration depth@15#. The relationships above arobtained by expanding at long times the properly normaliexpressions for the dynamic structure factors in DWS whcan be found in Ref.@15#. The lengthsl B and l T are theequivalent of the inverse scattering vectorq in the DLS ex-periments, and one hasl B! l T . Therefore, these two obsevation geometries allow us to probe relaxation phenomoccurring at two different length scales. From experimeperformed on latex suspensions in water, we obtainedl B52.3 nm andl T516 nm.

The sample was either translated by a PC-controlled smotor analogous to the one employed in the DLS setupwas displaced with an electroacoustic@21# device. In thesecond case the sample is translated at a frequencies be60 and 100 Hz, the amplitude of the oscillatory motion beon the order of 1 mm.

We used two different types of correlators. For the detmination of the dynamic structure factor, following the samprotocol as in DWS we used the ALV 5000 correlator. Fthe correlation echo technique we used a BI 9000 correlawhich can be configured in such a way as to concentragreat number of correlation channels around the delay ticoinciding with the peak positions, thus allowing their mesurement with a good precision@21#.

C. Experimental results and discussion

Figure 1~a! shows the dynamic structure factor of a gwhose cross-linking ratio is 231023. This is a typical case

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for which DLS and DWS measurements are complementFigure 1~a! displays theq variation of the dynamic structurefactor as measured by DLS. Whatever the magnitude ofwave vector transferq, the structure factor does not go tzero when time goes to infinity but reaches a nonzasymptotic value which depends on the value ofq. Follow-ing Ref. @14#, we took

limt→`

f ~q,t !5e2q2Dr 2~`!/65e2q2d2. ~18!

As illustrated in the inset of Fig. 1~a!, where the datacollected for cross-linking ratiosRc51.631023 and Rc55

FIG. 1. ~a! Dynamic structure factors for a gel~Rc5231023,Cp50.05 g/cm3! containing polysterene nanospheres~85 nm! at avolume fractionf51024. The different curves correspond to scatering angles of 35°~open circles!, 75° ~open squares!, 105° ~opendiamonds!, and 135°~open triangles!. The inset shows theq2 de-pendence of the limiting value of the dynamic structure factorlong times,f (q,`). Data sets correspond to gels having the saconcentration and volume fraction of particles as above; crolinking ratios are equal to 531023 ~circles!, 231023 ~squares!,and 1.631023 ~triangles!. Note that the values of they axis arelogarithmic. Solid lines are linear least-squares fits to the [email protected]. ~18!#. ~b! Dynamic structure factor of an equivalent samp~Rc5231023, and Cp50.05 g/cm3 and 107-nm particles atf52.531022! as measured by DWS in the transmitted~squares! andbackscattering~circles! geometries.

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31024 are also presented, the experimental results canfitted very satisfactorily with Eq.~17!, leading to the deter-mination of the characteristic spatial length explored byparticles,d, as a function of the cross-linking ratio~cf. TableI!. However, to better determine the dynamics of the binside this Brownian cage, one needs probe smaller lenscales. Figure 1~b! represents the dynamic structure factoobtained from the DWS intensity correlation functionssample similar to the one studied by DLS. We representhe measurements corresponding to the backscatteringtransmission geometries. First of all, the measured strucfactors go to zero at long times, both in the transmissionin the backscattering geometries, showing that the confiment of the particles cannot be seen at the scales probethis technique. The dynamics of the particles at short lenscales thus obtained shows a subdiffusive motion ofprobes. In all the samples we studied, the motion ofparticles was never purely diffusive, even at the shortesttection times.

We focus now on more strongly cross-linked systems,which nonergodicity effects are apparent even in DWS.performed series of autocorrelation experiments in whichsample was moved by steps of 200mm, thus observing dif-ferent pointsro of the speckle pattern. We first checked thgeneral assumptions of our treatment were valid, by plott@gr o

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2 versus^I & t2. Figure 2 shows the typica

curve obtained by plotting 2000 data points, each one cosponding to a different position of the sample. The datain very good agreement with the linear relationship@Eq.~17!#. This linear dependence was found in all the samppresenting a nonergodic behavior. Therefore, in this caseelectric field of the scattered intensity can be decompointo constant and Gaussian parts, and the time average ofluctuating part of the intensity does not depend on the pof observation.

FIG. 2. Experimental verification of the ‘‘self-heterodyne’’ relationship@cf. Eq. ~18!# in DWS ~backscattering geometry!. On thex axis are plotted the total scattering intensities~time-averaged!corresponding to a total of 2000 observation points, plotted ongraph. The sample is a gel withRc5131022, Cp50.05 g/cm3, andfo52.531022. The solid line is a least-squares linear fit to tdata; the slope is proportional to the fluctuating component ofscattered intensity@cf. Eq. ~17!#.

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In spite of the fact that the latex particles embedded ingels are strong light scatterers, in DLS experiments it ispossible to neglect the component of the intensity associwith the polymer concentration fluctuations of the bare gI PAA @22#.

The dynamic structure factors obtained by DLS wereted with a nonlinear least-squares fitting procedure usingexpression

f ~q,t !5 f ~q,`!1A exp~2t/tPAA!

1@~12A2 f ~q,`!#exp@2~ t/tc!g#. ~19!

This functional form was chosen primarily because it allofor quick, reproducible and satisfactory fits of the data,illustrated by Fig. 3~a!. The amplitude parameterA is theratio between the scattering intensity associated withconcentration fluctuations of the gel and the fluctuating pof the scattering intensity associated with the motion ofprobe particlesI F , i.e., A5I PAA /I F ; tc and tPAA are thecorrelation times associated with the particle motions andpolymer concentration fluctuations of the gel, respectiveAs illustrated in Fig. 3~b!, the values oftPAA measured in thedifferent gels are in good agreement with those measurean equivalent solution at the same concentration of the~crosses!, showing the usualq22 dependence@9,23–25#.

The values of the exponent of the stretched exponeng, reported in Table I, are essentially independent onscattering vector for a given system, but depend systemcally on the cross-linking ratio. The evolution ofg withcross-linking ratio could bear relevance to theoretical woon relaxation phenomena in disordered media. The evoluof the exponent could be related to the increasing amounfrozen-in disorder in the gel matrix surrounding the proparticles @26#. This ‘‘disorder’’ ought to be an increasingfunction of the cross-liking ratio, as the degrees of freedof the polymer concentration fluctuations are increasinhindered by the presence of physical links between neighing chains. In order to compare the characteristic timestccorresponding to different values ofg, we followed Delsantiand Munch in Ref.@27# and calculated the averaged tim^tc&5G(g)tc /g, whereG is the Eulerian gamma function.

The correlation timetc can be viewed as the characteritic exploration time of the Brownian cage, i.e., the time eaparticle will require to explore a significant portion of itallowed positions. This correlation time ought to be indepedent on theobservationscaleq21. Figure 3~b! clearly illus-trates this point for our experimental systems. The valued2 and^tc& as obtained from the above analysis are reporin Table I.

As the cross-linking ratio increases, the average msquare displacement decreases and eventually can no lobe detected reliably using DLS. The main motivation fusing DWS is then to expand the range of detectionsmaller displacements.

In Fig. 4 we show the dynamic structure factors obtainin the backscattering and transmission geometries, for crlinking ratios of 531023 @Fig. 4~a!# and 1022 @Fig. 4~b!#.We now see that the structure factor reaches a nonzero vat long times, from which we can deduce the maximumean square displacement of the particles, according to

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FIG. 3. ~a! Example of DLS dynamic structure [email protected], Cp50.05 g/cm3, andf

•5131024 ~85-nm latex spheres!,

and the scattering angle is 90°#. The decay function is bimodal. Thsolid line is a least-squares nonlinear fit to the data using Eq.~19!.The residual plot of the fit is shown on the bottom of the figuNote that the maximum deviation is less than 3%.~b! Variation ofthe characteristic decay times associated with the ‘‘cage’’ dynics, tc , and with the polymer concentration fluctuations of the gtp , as a function of the magnitude of the scattering vectorq. Filledsymbols correspond to thetc of gels with Cp50.05 g/cm3 at f51024 (85-nm polystyrene nanospheres!; cross-linking ratios are531023 ~diamonds!, 231023 ~squares!, and 1.631023 ~circles!.Open symbols show the evolution oftp for the corresponding gelsFor comparison, the decay time measured on a PAA solution asame concentration prepared in the same conditions~crosses! is alsoshown.

The values ofd2 obtained by using the above expressioare reported in the table. In both cases,d2 would have beentoo small to be measured reliably using DLS.

Let us remark that this way of performing explicitly thensemble averages is time consuming, and that at leastkey quantity, namely the plateau value of the mean squdisplacement, can be obtained more readily by computhe intensity correlation function while the sample is in ocillatory motion. During one period, the light from a largnumber of different points of the speckle is collected, andensemble average over all these points is performed intime. The shape of the correlation function reflects bothmacroscopic motion of the sample and the dynamics ofbeads themselves inside the gel. The filled circles in F4~a! and 4~b! show the results obtained in these experimenOne sees a first decrease due to the collective motion ofsample imposed by the translation, then a series of echcentered around the period of excitation~two echoes in thepresented experiment!, as the sample comes back periodcally to the position it occupied one period before. Duetheir random motion inside the gel matrix, the scatterers

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FIG. 4. ~a! Dynamic structure factor of a gel measured by DWin the transmitted geometry~open circles!. Sample characteristicsRc5131022, Cp50.05 g/cm3, andf

•52.531022. Filled circles

show the result of a correlation-echo experiment performed onsame sample.~b! DWS dynamic structure factors~open circles! andcorrelation echo~filled circles! for the same sample as in Fig. 3~a!,but now in the backscattering geometry.

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not come back exactly to their previous positions, andheight of the correlation echo is thus smaller than 1. In fawe recover the value of the ensemble average structure faat discrete times corresponding to integer multiples ofoscillatory frequency. Thus we are able to measure the mmum mean square displacement, provided that the periothe imposed translation is larger than the correlation timethe structure factor due to the motion of the beads arotheir average position.

Let us come back to the full structure factor. GeneDWS theory allows us to express it as a function of tensemble averaged mean square displacement^Dr 2(t)& @12#,from which we deduce, by a point to point numerical invesion of the formulas, the ensemble-averaged mean sqdisplacement of the probe particles. The results are giveFig. 5. At long times, one sees that the motion of the beadconstrained: they are trapped in Brownian cages whosedecreases when the cross-linking ratio increases. In Tabwe report the value ofd2, as obtained from the long-timbehavior of D r

2(t)& using Eqs.~20! or ~21! depending on thevalue ofd2. Also are given the values of^tc& determined byfitting ^D r

2(t)& to the equation@14#

^D r2~ t !&56d2@12exp@2~ t/tc!

g##, ~22!

At short times, the beads motion tends again toward a sdiffusive behavior reflecting short time dynamics of the gWe notice that, at short times, the beads in the less crlinked gel appear to undergo a slower local diffusion asmean squared displacement curve is the lowest.

It should be noted that, at present, the short time behaof beads trapped in a viscoelastic medium gives rise tdebate. For instance, the issue on whether one recoverdiffusive behavior of probes in water or not is not yet rsolved. In the present case, the experimental observatiolikely due to the fact that a weakly cross-linked gel hashigher fraction of dangling ends and highly branched s

FIG. 5. Mean square displacement as a function of timeprobe particles embedded in different gels matrices withCp

50.05 g/cm3 and fo52.531022; the cross-linking ratios of thedifferent samples areRc5231023 ~open and filled circles!, Rc

5531023 ~plus signs! and Rc5131022 ~squares!. The spatialranges accessible to DWS~backscattering! and DLS are shown onthe right hand side. Open circles correspond to DLS measurem

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parts that do not contribute to the elastic modulus butcrease the local viscosity experienced by the particles. Sond, we notice a slight shift between the measuremeperformed using DLS and DWS which cannot be accounfor by the slight difference of probe sizes. This is very likedue to the lack of reproducibility of the mechanical charateristics of the gel samples. Ideally, one would want to stuthe same gel with two very different concentrations of postyrene spheres. This is not possible, and we had to predifferent samples for each technique. Unfortunately, thgels are close enough to the gel point that small deviationthe sample preparation conditions~e.g., added cross-linking!can cause some differences in the mechanical propertiethe final sample.

We estimated the shear modulusm of the gel, as probedby the particles, from the value of the maximum mean squdisplacement, Dr 2(t)&, arguing that the work necessarymove a bead of radiusR by a distanceDr , inside a mediumof elasticity is of thermal origin@4# so that

m5kBT

6pRd2 , ~23!

kBT being the thermal energy. In fact, as discussedSchnurret al. @28#, the above expression is strictly valid onat time scales which are short enough so that the solventthe network move as one at scales large when comparedthe mesh size of the network. In this limit the gel behavesan incompressible fluid. An exact expression for longer tiscales should include the contribution due to the compsional deformation since the stress in the solvent has timrelax completely as the network is deformed. The correctfactor depends on the Poisson ratio and is negligible iflatter quantity is close to1

2. For polyacrylic acid gels pre-pared at a polymer concentration larger than those conered in this study@29#, and for polyacrylamide gels in goosolvent @30# the Poisson ratio was found to be close to1

3.With such value of the Poisson ratio the values ofm as de-termined by using Eq.~23! would be underestimated babout 12%, which is a small correction considering the lavariation ofm as a function of the cross-linking ratio showin Fig. 6.

When fitted to a power law, the elastic modulus variatiwith the cross-linking ratio yield an apparent scaling expnent of 2.9160.18. This result is not what one expects frothe classical theory of rubber elasticity; however, this behior, which has been observed previously, can be accoufor by the existence of dangling chains and unattached chat low cross-link density and of trapped entanglementshigh cross-link density. The former effect tends to decrethe shear modulus, the latter to increase it@31#. The discrep-ancy between the value of the local elastic modulus ofgel with Rc5531023 found using DLS and DWS mighbeexplained by the fact that the DLS measurement is cleat the very limit of the resolution of this technique@cf. theinset in Fig. 1~a!#. This is precisely the reason why we resorted to DWS for these gels. The two samples were issfrom different syntheses which introduce further uncertaiin the result.

We then compared the macroscopic measurement ofelastic modulus with optical measurements. We measu

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the elastic modulus of a gel cross-linked atRc5431023,with a Haake RS600 rheometer, and found that it didvary with frequency between 231022 and 2 Hz its valuebeing 20 Pa. The comparison with macroscopic measments is not straightforward. In fact, macroscopic measuments are usually performed by means of a uniaxial copression technique that provides the zero-frequemodulus. Measurements corresponding to the frequencymain accessible to the optical measurements are not poswith classical rheometers. A frequency-dependence ofstorage modulus of weakly cross-linked gels is oftenserved, and this could account for the difference betwmacroscopic and microscopic measurements. Moreomacroscopic dynamic measurements in cross-linked polygels are not straightforward, and are often marred by systatic errors which are very difficult to detect~e.g., surfacedefects and/or presence of a lubrication layer betweenhydrogel and the plates of the rheometer!, that lead to anunderestimated value of the shear modulus.

The characteristic time required by a particle to exploits Brownian cage tc& is independent on theq value, asshown in Fig. 3~b!, since it only depends on the charactertics of the sample and not on the scale of observation.possible to relate the average mean square displacemethe probe particlesd2 and ^tc& to deduce the local frictioncoefficient:

v5kBT^tc&6pRd2 . ~24!

Equation~24! is essentially a Stokes-Einstein relationshv having the dimensions of a viscosity~Pa s!. The values ofd2, ^tc&, v, andm for the different systems we studied asummarized in Table I.

FIG. 6. Evolution of the local shear modulusm as a function ofthe cross-linking ratioRc ~filled symbols!. The local friction coef-ficient v is also plotted~open symbols!.

t

e-e--yo-blee-nr,

er-

e

e

-isof

,

Contrary to the local shear modulusm, the friction coef-ficient defined above isindependenton the cross-linking ra-tio, as shown in Fig. 6. We note the satisfactory agreemenDWS and DLS results, especially considering the fact tsize of the particles used in the two different types of expements differed by about 20%. The local friction coefficientvreflects local relaxation processes of the polymer mawhich are not related to the cross-linking ratio of the pomer network. This result is not trivial and requires furthinvestigation.

The mean square displacements of the particles coulprinciple be further analyzed by Laplace transform inversof data to obtain the frequency-dependent storage andmoduli of the matrix@4#, which could be compared to macroscopic measurements. However, the numerical uncertties of this procedure pose a problem. Also, an actual coparison would require reliable macroscopic measuremethat are very difficult to obtain with classical rheometersthe same frequency range accessible to the optical meaments.

IV. CONCLUSION

We extended DWS to the case of nonergodic systems,used this technique to measure the motion of colloidal pticles trapped inside a chemically cross-linked gel. Theherent nongodicity of the scattering in these systems habe addressed experimentally by performing explicit esemble averages, or by using other averaging schemesmeasured the maximal amplitude of the motion of the pticles, which we considered to act as viscoelastic probethe gel matrix. From the mean square displacement weduced the variation of the elastic modulus of cross-linkgels. Further work has to be done to compare the optresults to macroscopic measurements performed in the sfrequency range. The results show that these scattering tniques are appropriate to investigate both dissipative stime properties and elastic moduli of permanently crolinked gels. To probe the short time behavior, one has tothe DWS, whereas the choice of the technique for measuthe elastic modulus will depend of the value of this modulUpon increasingG, one turns successively to DLS, thenDWS in the transmission mode, and eventually to DWSthe backscattering mode.

Finally, one should note that such techniques provide nprospects for studying inhomogeneities of materials or flowing the local mechanical response of time-evolving stems such as associating systems or gels under swellindeswelling.

ACKNOWLEDGMENTS

We thank F. Lequeux and J. F. Douglas for helpful coments on the paper, and D. J. Pine for fruitful discussion

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