Micro Gels

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Journal of Colloid and Interface Science 302 (2006) 537546www.elsevier.com/locate/jcis

Soft water-soluble microgel dispersions: Structure and rheology

A. Omari a,, R. Tabary b, D. Rousseau b, F. Leal Calderon a, J. Monteil a, G. Chauveteau b

a Laboratoire des Milieux Disperss Alimentaires, ISTAB, Avenue des Facults, 33405 Talence cedex, Franceb Institut Franais du Ptrole, 1-3 avenue de bois Prau, 92506 Rueil Malmaison cedex, France

Received 24 March 2006; accepted 6 July 2006

Available online 11 July 2006

AbstractThe size and structural characteristics of polyacrylamide-based water-soluble microgel dispersions were investigated by optical and rheological

methods. Microgel hydrodynamic radii R h were measured by light scattering and derived from intrinsic shear viscosity []0 . The variations of R 3h and []0 with the crosslink density N x , follow the scaling law R

3h= N x with close to 0.63, in good agreement with the simple structural

model proposed in this paper showing how the exact value of depends on inner structural details of the microgel. The plateau viscosity versusparticle apparent volume fraction shows a monotonous change from hard sphere dispersions (high crosslink density of microgels) to exible linearpolymer solutions. Measurements of the rst normal stress difference N 1 show that increasing the microgel crosslink density affects the systemviscosity more than its elasticity. Under oscillatory shear ow, loss and storage moduli undergo both qualitative and quantitative changes withcrosslink density. At moderate concentrations, the elastic modulus is the most affected and its slope in low frequency regime decreases from twoto less than one as N x increases. We discuss the experimental results within the frame of knowledge on linear, branched polymer solutions andsoft microgel suspensions. 2006 Elsevier Inc. All rights reserved.

Keywords: Microgel; Crosslink; Shear viscosity; Loss modulus; Storage modulus; First normal stress difference

1. Introduction

Micrometric and sub-micrometric particles are widely usedon an industrial scale and are usually designed to have suitableproperties depending on the target application [1]. For con-trolled drug delivery and, more recently, in Enhanced Oil Re-covery (EOR) processes, new soft particles, which we refer toas microgels, are very attractive [2]. For medical diagnosis, sur-face grafting of special molecules on rigid magnetic particles is

aimed at [3]. Different techniques have been developed to man-ufacture microgel particles. One of them consists in crosslink-ing a primary linear water-soluble polymer solution under con-trolled shear [4]. The emulsion polymerization technique ismore attractive since it leads very easily to well-calibrated par-ticles, which softness is mainly monitored through monomer-to-crosslinker concentration ratio [57]. In the past, signicantefforts were focused on investigating the swelling/de-swelling

* Corresponding author. Fax: +33 5 56 37 03 36. E-mail address: [email protected] (A. Omari).

mechanism that particles undergo by changing crosslink den-sity [8,9], temperature [10], particle charge density and/or sol-vent parameters such as pH [11] or salt content [9,12]. Forthat purpose, photon correlation spectroscopy (PCS) and small-angle neutron scattering (SANS) [1] are more suitable in di-lute regime and turbidimetry in higher concentrations [13].The particle swelling magnitude is then evaluated through themeasurement of Q = (R h/R h0)3 where Rh and Rh0 stand forthe hydrodynamic radius of particles in swelled and collapsed

states, respectively. As expected, R h is controlled by the parti-cle crosslink density N x . In a recent paper dealing with nano-colloidal particles, Q was found to vary as N x [9] where depends on monomer solvent quality. When used for EnhancedOil Recovery purposes, for instance in water shutoff and con-formance control operations, microgels adsorb onto the poresurface oil-bearing layers, reduce effective pore size and thuswater permeability. To be efcient, microgels must be cali-brated, highly water-soluble and soft enough for adsorbed layerthickness to be easily reduced by capillary pressure so that oilpermeability is not affected. In other terms, both particle size

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2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2006.07.006
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538 A. Omari et al. / Journal of Colloid and Interface Science 302 (2006) 537546

and their intrinsic mechanical characteristics must be speci-cally designed for the reservoir to be treated.

In most applications, these microgels are submitted to hy-drodynamic forces and their rheological behaviour should bewell understood and monitored. The main key parameters thatgovern such rheology are: particle surface charge, particle sizeand distribution, microgel phase volume fraction and particleinternal structure. Under shear ow, both the plateau viscosityand the shear-thinning behaviour deviate appreciably from thatof hard spheres [8]. Generally speaking, these dispersions areexpected to exhibit rheological behaviours ranging from that of hard sphere suspensions to that of linear polymer solutions witha determining role of crosslink density. So far, most publishedinvestigations dealt with highly crosslinked particles and wererestricted to linear properties under steady and oscillatory shearow. Moreover, little is known about the rheological propertiesof this type of suspensions in the non-linear regime.

In this paper, we propose to investigate the effects of change

of crosslink density on the structural and rheological proper-ties of well-characterizedmicrogel dispersions. A wide range of crosslink density was explored, from linear polymer to highlycrosslinked microgels. Furthermore, the concentration range issuch that both dilute and concentrated regimes were explored.In steady shear experiments, viscosity and rst normal stressdifference were measured in both linear and non-linear regimes.Dispersions were also submitted to oscillatory shear ow, andmeasurements of loss and storage moduli were performed in thelinear regime, within a frequency range accessible to the equip-ment used.

2. Experimental

The particles used in this study were acrylamide-based mi-crogels kindly supplied by SEPPIC Company (France). Theywere chemically similar to those previously used by Senff andRichtering [10]. They were obtained by using emulsion poly-merization technique, which generally leads to a narrow sizedistribution. The monomer concentration in emulsion dropletswas kept constant, and the particle internal crosslink densitywas monitored by varying the amount of added crosslinker. Asthe emphasis here is on general features, the detailed chem-istry of particles is not relevant and is not described. A linearpolymer with similar chemical composition was also used. The

solvent was brine containing 20 g l 1

NaCl and 0.4 g l 1

NaN3(used as a bactericide and as a stabilizer to prevent free radi-cal degradation), both dissolved in distilled water. The pH wasadjusted to 7 for all samples.

Microgel sizes were measured by the dynamic light scatter-ing technique (Photon Correlation Spectroscopy), which allowsthe determination of the hydrodynamic radius R h of dispersedparticles through their translational diffusion coefcient D 0:

Rh = kBT / 6 sD 0 ,

where kB is the Boltzmanns constant, T is the absolute tem-perature and s is the solvent viscosity. Experiments were doneat sufciently low angles in order to meet the conditionqR h < 1, q being the scattering vector [q = (4n/) sin(/ 2)]

where n is the solvent refractive index and the wavelength of the laser source ( = 632.8 nm). The dispersions used were suf-ciently diluted ( < 200 ppm) to prevent multiple scattering andinteraction effects. The experiments were performed using theN4 Plus 3000 apparatus from Beckman Coulter at 3 differentangles (15.7 , 23 and 30.1 ).

Further characterizations were then carried out by measur-ing the viscosity for very dilute dispersions using a low shearviscometer (LS 30 from Contraves) equipped with a Couettegeometry (0.5 mm gap) at shear rates ranging from 0.1 to70 s 1 . In this way, it was possible to accurately determine therelative viscosity r0 , dened as the plateau viscosity ratio be-tween the dispersion and the solvent.

For higher concentrations, a controlled-stress rheometer(MCR 300 from Anton-Paar Physica) and a controlled-strainrheometer (ARES from Rheometric Scientic) were used, bothequipped with coaxial cylinders or cone and plate geometry.These instruments enabled us to measure the steady shear vis-

cosity in linear (if it does exist) and in non linear regimes.The rst normal stress difference, N 1 , and the dynamic moduliversus frequency in oscillatory shear linear regime were alsomeasured.

All measurements were performed at 20 C unless otherwisespecied.

3. Results and discussion

3.1. Microgel size determination

Microgels with a high crosslink density were rst examinedwith an optical microscope, which qualitatively reveals that par-

ticles are spherical and closely monodisperse. The crosslink density N x is dened as the ratio of crosslinker-to-monomerconcentrations, assuming that both of them are homogenouslydistributed. The value of N x ranges from 0 to 0.3%; zero cor-responding to linear polymer. In Fig. 1, particle radii measuredby dynamic light scattering were plotted versus N x . Althoughthe results are scattered, this gure shows that microgel radiusdecreases with crosslink density according to a scaling law:

(1)R 3h = N x

with close to 0.63. For microgels obtained by emulsion poly-merization of methacrylic acid and ethyl acrylate cross-linkedwith diallyl phthalate, Tan et al. [9] found a larger power lawexponent ( 1.5).

In the model we propose, microgel size and structure derivefrom well known scaling laws [14] as follows.

If we consider a microgel as made of n b subunits linked bythe covalent bonds formed by the crosslinker, which we willrefer to as blobs, characterized by a mass m and a distance between two consecutive crosslinks, the microgel radius of gy-ration Rg is given by:

(2)R 3g = n b 3.

As a rst approximation, and m should be related by a MarkHouwink-type relationship:

(3) = m ,


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A. Omari et al. / Journal of Colloid and Interface Science 302 (2006) 537546 539

Fig. 1. Plot of hydrodynamic radius ( Q) and particle intrinsic viscosity ( F ) of microgel suspensions as a function of crosslink density. ( ) []0 = (N x ) 0.63 ;(--) R 3h

= (N x ) 0.63 .

where the exponent is the excluded volume parameter, thevalues of which depend on chain stiffness and solvent quality.For exible chains, we expect values to be 0.5 and 0.6, formonomers in and good solvent respectively, while for fullystiff chains = 1.

The microgel mass M is related to m by:

(4)M = mN x .

Combining Eqs. (2) to (4) gives:

(5)R 3g = M 3 N (1 3)x .

For monodisperse microgels, M is constant so that R 3g is relatedto N x by a scaling law:

(6)R 3g = N x

with = 3 1, so we can predict = 0.5, 0.8 and 2, forexible chains in solvent, exible chains in good solvent andfor fully stiff chains, respectively.

This simple model has predictive capability, as soon asthe chain chemistry and solvent quality are known. In thecase of our experiments, with the chemistry of microgels usedand the experimental conditions (temperature and water salin-

ity), our model predicts an value between 0.5 and 0.8 ingood agreement with experimental results ( = 0.63), since themean monomer number between successive crosslinks is smallenough for volume exclusion effects to be limited. In the case of the experimental results reported in Ref. [9] where was foundto be equal to 1.5, the above model suggests that the polymerchain is rigid, in agreement with polymer chemistry and wa-ter salinity. Indeed, such exponent was observed for free saltmedium and for neutralisation degree of polymer acidic groupsexceeding 0.4 (signicantly charged microgels).

In the concentrated regime, the radius of soft microgels is ex-pected to be reduced because of its compressibility. However,more sophisticated techniques are needed to determine the par-ticle radius in that regime.

3.2. Shear viscosity measurement

Relative shear viscosity measurements were performed oneach sample at variable microgel concentrations, C . At lowshear rates and low concentrations, the reduced specic viscos-ity spr is expressed by:

spr = ( r0 1)/C,

where spr is a linear function of C , r0 being the relativeplateau viscosity. From the spr(C) plot, the microgel intrinsicviscosity []0 is determined by extrapolation at C = 0, and the

Huggins constantK

H from the slope. []

0 andK

H , respectively,represent the hydrodynamic volume occupied by an isolatedparticle and the particleparticle interactions. Table 1 summa-rizes the results obtained for most microgel samples used, andfor the linear polymer. On the one hand, we observe an increasein the Huggins constant with crosslink density from 0.2 for lin-ear polymer to 1 for the highest N x value. Even for highlycrosslinked microgel dispersions, KH = 1 as expected for re-pulsive hard spheres without exceeding this value, thus showingthe absence of attractive interactions between microgels. On theother hand, microgel intrinsic viscosity (which is proportionalto particle volume) decreases when N x increases. Intrinsic vis-cosity values are also reported in Fig. 1, showing that both []0and R 3h exhibit the same scaling law versus N x . For linear poly-mers of molecular weight M w, []0 is linked to the radius of gyration R g through the relationship: []0 = R 3g /M w, where is a universal constant. The absolute values of intrinsic vis-cosities are signicantly larger (nearly one order of magnitudelarger) than that previously studied in literature. The microgelsused in this study can thus be considered to be soft particles.

As it is customary to do when dealing with microgels, ex-perimental results are more conveniently expressed in termsof apparent particle volume fraction , rather than in massconcentration [8,15,16] . This is achieved in dilute regime bymatching the linear terms in Huggins and Einstein equations:

(7)r0 = 1 + C []0 + K HC 2[]20,


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Table 1Microgel characteristics

Samplenumber

Crosslink density,N x (%)

[]0(cm3 g 1)

K H Rh(m)

1 0 2840 284 0 .26 0.03 0.20 0.022 0.05 1335 134 0 .21 0.02 1.60 0.16

3 0.10 884 88 0.40 0.04 1.40 0.144 0.15 670 67 1.33 0.135 0.20 559 56 1.40 0.146 0.30 428 43 1.0 0.1 1.15 0.12

(8)r0 = 1 + 2.5.

Thus giving: = 2C []0/ 5.For high concentrations (i.e., high volume fractions), we

measured shear viscosities versus shear rate. Fig. 2 shows thevariation of the low-shear relative viscosity as a function of for the six samples. For some of them, no plateau viscosity wasobserved at high volume fractions, and the data between brack-

ets corresponds to the lowest accessible shear rate.For estimation of microgels softness (deviation from hardsphere suspensions), it is convenient to use the empiricalKriegerDougherty equation:

r0 = (1 / m) 2.5 m .

The curve corresponding to hard spheres is plotted on the samegure for comparison. In this relation, the maximum random-packing fraction m is taken as being equal to 0.64 (monodis-perse spheres). It is to be noted that, for general use, the quantity2.5 m should be replaced by []0m . As pointed out earlier,microgel dispersions exhibit lower dependence of r0 on thanhard sphere dispersions [8], but signicantly higher than linearpolymer [2]. Such dependence becomes weaker as crosslink density decreases, owing to partial particle inter-penetrability.The range of such inter-penetrability is expected to be of theorder of the blob mesh size of the internal network. The ac-tual conformation of the microgels in concentrated regime isthe result of two opposite factors: particle inter-penetrability,given by shell properties, and compressibility, governed by core

crosslink density. At low N x values, interpenetration is possibleover large scales, and dominates both structure and rheology insemi-dilute regimes, so that the behaviour is close to that of linear polymers. By contrast, at large N x values, microgels aremore and more expected to behave like hard spheres.

When particle inter-penetration can be neglected (highcrosslink density), one can dene an effective volume fraction,eff , by the Batchelor equation [10,13]:

r0 = 1 + 2.5 eff + 5.9 2eff .

The ratio of the apparent volume fraction to the effectivevolume fraction eff is a useful parameter, since it is repre-sentative of particle compressibility. Nevertheless, since onlyrst order and quadratic terms are involved, the data does nott at high concentrations. However, such an approach is notpertinent in the present study, for at least two reasons: ourmicrogels are very soft, and can interpenetrate substantially,and concentrations used are too high. Nevertheless, one should

keep in mind that imbedding a particle in an imposed shearow causes an extra shear rate by modifying the ow pat-tern [17]. Such shear rate excess is, of course, a decreasingfunction of the particles inner permeability. In another picture,the shared zone of neighbouring particles (increasing with shellthickness) decreases as crosslink density increases. As a conse-quence, the local shear rate becomes higher, and the local shearstress rises. Conversely, decreasing crosslink density inducessignicant particle interpenetration, thus weakening the depen-dence of r0 on . Roovers [18] has reported similar behaviourfor concentrated star polymer solutions when the number of branches per molecule increases. For linear polymer, as indi-cated by the dashed line in Fig. 2, the relative viscosity closelyfollows the power law, r0 = 3.4 , which is known to t exper-imental data correctly in the high concentration regime [19]. Asexpected, the rate of viscosity rising for microgel suspensionsincreases monotonically with N x .

In the high shear rate regime, as shown in Fig. 3, shear-thinning index n increases signicantly with . Such shear-thinning index n is the exponent in the pseudo-plastic regime,

Fig. 2. Variation of relative plateau shear viscosity with apparent volume fraction for different crosslink densities: ( ! ) 0%; (E ) 0.05%; (P ) 0.10%; ( 1 ) 0.15%;(+ ) 0.20%; (| ) 0.30%; ( ) KriegerDougherty relation; ( ) correspond to the power law: r0 = 3.4 .


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Fig. 3. Shear-thinning index versus apparent volume fraction for various N x . (" ) 0% (linear polymer); ( F ) 0.05%; ( Q) 0.10%; ( + ) 0.20%; (| ) 0.30%.

Fig. 4. Normalized relative shear viscosity versus shear rate at = 0.34 for lin-ear polymer ( N x = 0%) and a microgel suspension ( N x = 0.3%), temperatureis 30 C. The continuous lines are data ts using the Carreau model.

where viscosity decreases according to the power law: r = n . For linear polymer, n rises smoothly and should laterreach a limiting value close to 0.8, which characterizes concen-trated high-molecular-weight linear polymers [19]. For micro-gel dispersions, the asymptotic value obviously increases withcrosslink density, due to the increase of local connement of peripheral microgels shell.

Furthermore, an isolated particle with low crosslink density

should deform easily under shear ow, for volume fractionsfar below m . Thus, increasing particle internal density is ex-pected to reduce shear-thinning behaviour. The results shown inFig. 4 conrm that, in dilute regime ( < m), shear thinningis due to individual particle deformation under shear forces,even if a larger shear rate interval should be scanned to con-rm this nding. By contrast, for > m , the shear-thinningindex increases with crosslink density. Moreover, the shape of the n vs curves is noticeably modied as N x varies, butthe asymptotic limit is somewhat similar. Such behaviour is in-debted to the intrinsic structure of the system. When particlesare highly penetrable (low N x ), increasing the system concen-tration results in an homogeneous increase of concentration.For partially penetrable particles (higher N x ), the resulting in-

crease in concentration is located in the inter-particles sharedzone, where viscous dissipation is predominant. Therefore theobserved shear thinning is more pronounced.

3.3. Measurement of rst normal stress differences

First normal stress difference N 1 was measured as a functionof the macroscopic applied shear rate using a cone and plategeometry. Most measurements were carried out in the high ap-parent volume fraction range, because data at low was tooscattered. They were also restricted to shear rates high enoughto ensure acceptable measurement accuracy, but neverthelessnot too high, in order to prevent ow instabilities [20,21] . Inpolymers, normal stressdifference in shear ow results from theanisotropic alignment of molecular segments. It depends there-fore on the tension along streamlines. Fig. 5a gives the varia-tion of N 1 with for each sample at apparent volume fraction = 5.75. It shows that suspensions of more dense microgelsclearly exhibit higher elastic characteristics, when compared atthe same imposed shear rate. To explain such behaviour, we em-phasize again that, instead of the imposed shear rate, one shoulduse the effective shear rate that segments actually experience inthe overlap region. In that region, tightening of ow stream-lines and consequently, the increase of local shear rate is more

important as the inner density of microgels is increased. More-over, in order to have an overall view of the incidence of struc-tural change on viscoelasticity under permanent shear ow, itis more convenient to represent N 1 versus shear stress [22]. Bydoing so, the hierarchy is inverted ( Fig. 5b) and N 1 is a de-creasing function of N x at a xed shear stress, meaning thatthe elastic-to-viscous force ratio is a decreasing function of mi-crogel crosslink density. In other words, increasing the internaldensity of microgels affects viscosity more than elasticity forthe same . This is more clearly evidenced when the micro-gel volume fraction is increased further, as shown in Fig. 5c. Atthat volume fraction, it was unfortunately not possible to ob-tain a homogeneous preparation for microgels with the highestdensity.


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(a)

(b)

(c)

Fig. 5. First normal stress difference for = 5.75: (a) versus shear rate and (b) versus shear stress. (c) First normal stress difference versus shear stress for = 7.

3.4. Oscillatory shear measurement

In this section, we report measurements of both loss andstorage modulus ( G and G ) as a function of frequency, un-der small amplitude oscillatory shear. All samples were tested

but only some results are presented for clarity. For the linearpolymer ( Fig. 6a), we observe the expected behaviour. Poly-mer solutions areessentially viscous at low frequencies, tendingto t the scaling laws: G = 2 and G = . At high fre-quencies, elasticity dominates ( G > G ). This corresponds to


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(a)

(b)

(c)

Fig. 6. Loss modulus (solid symbols) and storage modulus (hallow symbols) versus frequency at indicated apparent volume for linear polymer (a) and for microgelsamples: (b) N x = 0.10%; (c) N x = 0.30%.

a Maxwell-type behaviour with a single relaxation time thatmay be determined from the crossover point and, as expected,this relaxation time increases with concentration. Therefore mi-crogel dispersions display a similar liquid-like behaviour atlow volume fractions but deviate from this behaviour at high values. Moreover, for identical apparent volume fractions,

the deviation widens out when particles internal density in-creases. For instance, at intermediate crosslink densities andlow volume fractions, the behaviour is similar to that of lin-ear polymer, except in the low frequency regime, where theexponent corresponding to the storage modulus is severely re-duced to less than one. In the low frequency regime, G and G


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(a)

(b)

(c)

Fig. 7. Plot of tan () versus frequency at indicated apparent volume fraction: (a) linear polymer; (b) N x = 0.10%; (c) N x = 0.30%.

change in the same way, with a subsequent change in the Gslope below the crossover point ( Fig. 6b). In the picture pro-posed above, soft microgels are described as deformable (soft)core-shell particles. Their elasticity at high is attributed tothe shell constituted by the dangling polymer branches lo-cated at the periphery, while elasticity at low reects the slowdynamics of cores. Similar behaviour was reported earlier by

Pham et al. [23] for latex dispersions containing associativecopolymers, with either attractive or repulsive particleparticleinteractions. These authors also proposed a simple model to ex-tract contributions from hard spheres and polymer solutions.Unfortunately, because of microgel compressibility and inter-penetrability, such a method cannot be used for quantitativetreatment of our data.


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Fig. 8. Plot of G values at frequency of 1 rads 1 , as a function of apparent volume fraction for various N x . (" ) Linear polymer; ( F ) 0.05%; (Q) 0.10%; ( 2 ) 0.15%;(+ ) 0.20%; (| ) 0.30%. (The dashed lines are only guides for eyes.)

For higher crosslink densities and high volume fractions,

microgel dispersions exhibit a power-law dependence of theviscoelastic moduli, with at intermediate frequencies (seeFig. 6c for example) where we have G = kG = p . Thisbehaviour reveals the existence of ordered structures in the dis-persions and the exact value of the exponent p is dependent onstructural details. In that frequency range, their viscoelasticityis therefore governed by a relaxation time spectrum of the form:

H() = H 0(/ 0) p ,

where H 0 (Pa) and 0 (s) are scaling factors [24]. This is moreclearly shown in Figs. 7a7c , where the viscous-to-elastic mod-ulus ratio (or the dissipated to stored energy ratio): k = tan() =tan(p/ 2) is plotted versus frequency for some samples at var-ious . For the linear polymer, the behaviour is usual inasmuchas k is a monotonically decreasing function of frequency and islowered by increasing polymer concentration. The correspond-ing p value is greater than 0.5 at intermediate frequencies, sincepolymer solutions are more viscous than elastic; G > G . Forall microgel samples, deviation from linear polymer behaviouris both qualitative and quantitative. First, no asymptotic diver-gence in the low frequency regime is observed, and secondlytan() is quasi-insensitive to frequency changes, especially forhigh volume fractions and high crosslink densities.

For low crosslink density (Fig. 7b), the viscous-to-elasticmodulus ratio is greater than 1 (giving 0 .5 < p < 1 since

p = 2/ tan

1(G /G )) in the intermediate frequency range.This indicates that such dispersions behave as very soft gels orsemi-dilute solutions of branched polymers. The exponent issimilar to that obtained by crosslinking exible linear polymersbelow the gel point with a lack of crosslinking agent [22]. Asinternal crosslink density increases, particles steric connementincreases and suspensions take a gel-like structure with p < 0.5at signicant values (see Fig. 7c). Indeed, for a solution-likesystem, the viscous part dominates, giving p > 0.5. For gelledsystems however, the elastic part dominates, and p is then lessthan 0.5. At the transition from solution-like state to gel-likestate (the gel point), mechanical and structural properties of thesystem are described by scaling laws which may be derived us-ing percolation theory. In that regime, both G and G scales

as p where the exponent p is constant over a large frequency

interval and which exact value is dependent on the involved sys-tem.In other respects, since for microgel dispersions elasticity

arises from particleparticle contacts, a critical volume frac-tion c can be dened, below which G should drop to verylow values. For > c, G rises sharply before reaching apseudo-plateau. Moreover, as the distance between neighbour-ing particles scales as 1/ 3 , G may be written in the form:

G = G p 1 (/ c) 1/ 3 ,

where G p represents the intrinsic elasticity of microgels and cis usually taken equal to m . Some authors have already usedthis relation in an attempt to plot data on a master curve ex-perimental data obtained for various crosslink densities [25].For that purpose, G p was taken as equal to the plateau modulusof the bulk gel at the considered crosslink concentration. Thedata did not t with a master curve, probably due to particlecompressibility, and adjustment parameters were necessary tomake them collapse. Therefore, and since in the present studycrosslink densities are an order of magnitude lower, we did notperform any scaling of our data and only qualitative argumentsare put forward to explain the observed behaviour. Fig. 8 showsthe evolution of the elastic shear modulus G ( = 1 rad s 1)as a function of . For low crosslink densities, G is a growingfunction of , as for linear polymers, since particles may easily

entangle without signicant compression. Further increase incrosslink density obviously reduces the critical volume fraction,and levels off the elasticity at high volume fractions, approach-ing the behaviour of dense microgel suspensions.

4. Conclusion

Microgels structure and rheological properties of microgelsuspensions were studied with a particular emphasis on the in-uence of internal crosslink density. Crosslink densities werein the low range, allowing particles to partially interpenetrate.We proposed a simple model that was able to predict the depen-dence of hydrodynamic radius of microgel particles on their in-ner crosslink density, at least qualitatively. Rheological behav-


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iour of studied Microgel suspensions is shown to lie betweenthat of linear polymer solutions and that of hard sphere suspen-sions with a smooth change as crosslink density increases. Inthis work, we only propose a phenomenological interpretationof experimental evidences and more efforts are to be undertakento establish quantitative modelling of the rheology of such sys-tems.

Acknowledgments

The authors wish to thank IFP for nancial support andSEPPIC Company for supplying microgels.

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