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An effective geometrical approach to the structure of colloidal suspensions of very anisometric

particles

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2010 EPL 90 36005

(http://iopscience.iop.org/0295-5075/90/3/36005)

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May 2010

EPL, 90 (2010) 36005 www.epljournal.org

doi: 10.1209/0295-5075/90/36005

An effective geometrical approach to the structure of colloidalsuspensions of very anisometric particles

C. Baravian1(a), L. J. Michot

2, E. Paineau

2, I. Bihannic

2, P. Davidson

3, M. Imperor-Clerc

3, E. Belamie

4

and P. Levitz5

1 LEMTA, Nancy University, CNRS, UMR 7563 - BP 160, 54501 Vandoeuvre Cedex, France, EU2 LEM, Nancy University, CNRS UMR 7569 - BP 40, 54501 Vandoeuvre Cedex, France, EU3 LPS, Paris Sud, CNRS, UMR 8502 - Batiment 510, 91405 Orsay Cedex, France, EU4 ICG, Universite Montpellier, CNRS, UMR 5253 - CC 1700, Montpellier Cedex 5, France, EU5 LPMC, Ecole Polytechnique, CNRS, UMR 7643 - 91128 Palaiseau Cedex, France, EU

received 6 March 2010; accepted 23 April 2010published online 28 May 2010

PACS 61.20.Gy – Theory and models of liquid structurePACS 61.05.cf – X-ray scattering (including small-angle scattering)

Abstract – We show in the present letter that the organization of colloidal suspensions of veryanisometric repulsive particles can be understood on the basis of simple geometrical considerations.Using a large set of rod-like and plate-like particles, we first evidence that the experimental inter-particle distances can be accurately predicted from geometrical constraints. We then show that theexperimental static structure factors can be satisfactorily fitted using an effective Percus-Yevickstructure factor. The fit parameters are then interpreted in terms of the co-excluded volumesof effective ghost particles, which further supports the geometrical representation previouslydeveloped.

Copyright c© EPLA, 2010

Due to their rich and complex phase diagrams andphysical properties, the organization of suspensions ofanisometric particles remains the object of intensiveresearch [1–7]. Of particular interest is their behaviorunder the influence of external fields and the relationshipbetween microscopic organization and macroscopic prop-erties. In this context, the use of a geometrical effectiveapproach to assess the influence of flow on structureappears particularly relevant, as already shown in thefield of suspension rheology [8,9] where it has been appliedmainly to hard or interacting spheres. In this letter, weextend such a geometrical approach to the case of veryanisometric colloidal repulsive rod-like and plate-likeparticles. We show that such an approach, mainly basedon excluded-volume interactions, can be applied in thewhole phase diagram to predict the average distancesbetween particles. In addition, to check the relevance ofthis geometrical procedure, we first model, by a calcula-tion in direct space, the experimental structure factorsobtained for various suspensions of anisometric particles,using an effective Percus-Yevick approximation of theliquid-structure factor [10,11]. We then interpret the

(a)E-mail: christophe.baravian@ensem.inpl-nancy.fr

values of the equivalent Percus-Yevick volume fractionson the basis of our geometrical approach.In order to test the robustness of our method, a wide

range of particle anisometries, from rod-like to plate-like,were investigated, using available literature data. Threerod-like particles were used: fd-virus [12], imogolite [13],and chitin [14], providing aspect ratios ranging from11 to 140. Size-selected natural clay samples were usedas models of plate-like particles. Beidellite SBd parti-cles [15] are disk-like whereas nontronites NAu1 [16] andNAu2 [7, 17] are lath-shaped. In this case, aspect ratiosrange from 150 to 400. Table 1 summarizes the dimensionsof the particles used in this study. t is either the thickness(for plate-like particles) or the diameter (for rod-likeparticles); a is the length and b the width of lath-shapedparticles, and L is either the length for rod-like particles orthe diameter for disk-like particles, or (a+ b)/2 forlath-shaped particles.Experimental structure factors were obtained from

either small-angle X-ray scattering [7,13–16] or small-angle light scattering [12] intensities that were dividedby the particle form factor (q−1 for rod-like parti-cles and q−2 for plate-like particles, where q is thescattering vector modulus, in the intermediate regime2π/L< q < 2π/t [17]).

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C. Baravian et al.

Table 1: Dimensions of the particles used in this study (innanometers).

System Morphology a b t Lfd-virus rod 6 880imogolite rod 3.2 130chitin rod 23 260SBd S1 disk 0.7 280SBd S2 disk 1.1 210Nau1 S1 lath 350 50 0.65 200Nau1 S2 lath 200 40 0.60 120Nau2 S1 lath 700 140 1.1 420Nau2 S2 lath 370 90 0.75 230Nau2 S3 lath 150 50 0.65 100

In order to derive the average inter-particle distance〈D〉 for various particle volume fractions, φparticle, we firstintroduce the volume fraction of spheres encompassing theparticles, φsphere, that can be written as

φsphere = φparticle(π/6)L3

particle volume;

– for rods, φsphere = φparticle(π/6)L3

(π/4)t2L= φparticle

2

3(L/t)

2;

– for disks, φsphere = φparticle(π/6)L3

(π/4)tL2= φparticle

2

3L/t.

For φsphere much smaller than a given critical value φ∗ (for

a face-centered cubic packing, φ∗ ≈ 0.74), the particles canfreely rotate. The isotropic center-to-center inter-particledistance is then given, as for spherical particles, by

〈D〉isotropic /L= (φ∗/φsphere)1/3 .

For φsphere of the order of or larger than φ∗, anisotropic

ordering occurs both in terms of inter-particle distancesand particle orientations. At large enough volume frac-tions, all systems display an isotropic-nematic phase tran-sition. We simply consider here that upon increasingconcentration, above φ∗, the spheres built around theparticles are progressively deformed along the largestparticle dimension until a dense packing of ellipsoids isreached. For rod-like (respectively, disk-like) particles, thisprocess leads to prolate (respectively, oblate) ellipsoids(fig. 1). Here, the virtual ellipsoids built around the parti-cles physically correspond to the average free volume avail-able per particle. For a given particle morphology andvolume fraction, this virtual volume will therefore growuntil it occupies the whole space available leading to avolume fraction of virtual particles equal to 1. For thisreason, in the present letter, we chose to use φ∗ = 1.For φsphere much larger than φ

∗, i.e. in the nematicphase, the smaller size of the virtual ellipsoids is equal

Fig. 1: (Colour on-line) Schematic representation of the ellip-soids built around the particles. Top: disks. Bottom: rods.

to the average distances between particles. Therefore, inthis regime, we obtain

– for rods, 〈D〉prolateorganized /L= (φ∗/φsphere)

1/2;

– for disks, 〈D〉oblateorganized /L= φ∗/φsphere.

A continuous description of the average inter-particledistance, from the dilute to the concentrated regime, cansimply be obtained by making the heuristic assumption ofa harmonic average between the two low and high sphericalvolume fraction limits:

1/ 〈D〉= 1/ 〈D〉isotropic+1/ 〈D〉organized .Equations (1) then provide the average inter-particledistance for rods (eq. (1a)) and for disks (eq. (1b)):

〈D〉rodL

=1(

φsphereφ∗

)1/3+(φsphereφ∗

)1/2 , (1a)

〈D〉diskL

=1(

φsphereφ∗

)1/3+(φsphereφ∗

) . (1b)

Figure 2 displays the comparison between the distancesobtained from eqs. (1) (with φ∗ = 1) and experimentaldata for rod-like particles (figs. 2(a) and (b)) and for plate-like particles (figs. 2(c) and (d)). For all experimentaldata, the inter-particle distance was deduced from theposition of the first correlation peak observed in theexperimental structure factors. A striking agreement isobtained whatever the organization of the suspensionsfrom the dilute isotropic liquid regime up to concentratedregions of the phase diagrams where nematic sols and gelsare observed.Considering suspensions of strongly anisometric parti-

cles as packings of effective ellipsoids, with aspect ratiosdescribed by eqs. (1), is clearly very efficient for determin-ing inter-particle distances.In principle, such an approach could allow the direct

calculation of the structure factor of the suspensions.Indeed, Donev et al. [18] recently proved that the 3Dorganization of densely packed ellipsoidal particles could

36005-p2

Structure factor of very anisometric particle suspensions

Fig. 2: Comparison between experimental and calculated inter-particle distances. (a), (b): rod-like particles. (c), (d): disk-likeparticles.

be univocally determined by direct calculation. Suchan exercise is, however, presently limited to thecase of moderately anisometric ellipsoids and cannotbe applied to the strongly anisometric particles investi-gated here.For this reason, in the present work, we test an alter-

native approach based on the use of Percus-Yevick equiv-alent spheres whose radius and volume fraction must bedetermined. A similar approach was recently applied tothe case of globular particles [11], the diameter 2aPY ofthe Percus-Yevick equivalent sphere being chosen as thesmallest size of the prolate or oblate ellipsoid. In the case ofvery anisometric particles, such as those investigated here,as 2aPY ≈ 〈D〉, the only unknown parameter is the equiv-alent volume fraction 〈φPY〉 that can then be adjusted toreproduce the experimental structure factors. As shown infig. 3, this treatment provides very satisfactory fits of theexperimental data for both rod-like and disk-like particles,which may seem quite surprising in view of the very largeparticle aspect ratios. This is most probably due to corre-lations between excluded volumes that are much largerthan the actual particle volumes.In the following sections, we propose a tentative geomet-

rical explanation of the validity of the Percus-Yevick treat-ment, based on the steric interactions between effectiveparticles. Once the effective particles are properly defined,the inter-particle contribution to the structure factor,in direct space, can be obtained from the determina-tion of the co-excluded volume occupied by the effectiveparticles [11,19].The first step in such an approach is to estimate the

sweeping volume of the effective particles defined as thevolume displaced upon one rotation involving a change

Fig. 3: (Colour on-line) Fits of the Percus-Yevick structurefactors for various particle volume fractions φ. (a) Sbd S1;(b) Sbd S2; (c) fd-virus; (d) imogolite. The continuous linesare results from modeling calculated as in ref. [20].

in the particle director. Such sweeping volumes can becompared with the excluded volumes defined by Onsagerfor rods and disks [1]. In the case of rods, the sweep-ing volume is that of a disk of diameter L and thick-ness t: V sweeprod = (π/4)L2t, i.e. the same value as thatobtained by Onsager (eq. (49) in [1]). In the case ofdisks, the sweeping volume is that of a sphere of diameterL: V sweepdisk = (π/6)L3, whereas Onsager obtains (π/4)2L3

(eq. (50) in [1]), i.e. a 15% difference.As shown above in the analysis of inter-particle

distances, considering the free volumes of rods and disksas virtual ellipsoids, of anisotropy increasing with particlevolume fraction, is relevant to describe such suspensionsof very anisometric particles. We are then led to calculatesweeping volumes for ellipsoids. For oblate ones, i.e. fordisk-like particles, the sweeping volume is the sphereencompassing the oblate ellipsoid, whose diameter (L) isthat of the particle (fig. 1), V sweepoblate =

π6L3.

The case of prolate ellipsoids, corresponding to rod-like particles is more complex. Indeed, in that case, thesweeping volume is given by an oblate ellipsoid of diameterL and thickness 〈D〉rod: V sweepprolate =

π6L2〈D〉rod. The radius

Req of the sphere having the same sweeping volume as theprolate ellipsoid is obtained from

4

3πR3eq = V

sweepprolate =

π

6L2 〈D〉 rod.

The intersection volume, Vco-volume, of two spheres ofradius R separated by a distance 〈D〉 is given by

Vco-volume =π

12(4R+ 〈D〉) (2R−〈D〉)2 .

36005-p3

C. Baravian et al.

Fig. 4: Percus-Yevick volume fraction vs. (a) the sphericalvolume fraction and (b) the reduced average inter-particuledistance. In (a), the dashed and continuous lines correspondto eqs. (3) and (4), respectively. In (b), the continuous linecorresponds to eq. (2).

The co-sweeping volume per unit sphere 〈φ〉CS is thereforegiven by

〈φ〉CS = 12

Vco-volume43πR

3=1

4

(2+〈D〉2R

)(1− 〈D〉2R

)2. (2)

In the case of disk-like particles, taking R=L/2, thefollowing expression then provides the co-sweeping volumefraction:

〈φ〉CSdisk =1

4

(2+〈D〉diskL

)(1− 〈D〉disk

L

)2. (3)

In the case of rod-like particles, introducing Req intoeq. (2), we obtain

〈φ〉CSrod =1

4

(2+

( 〈D〉rodL

)2/3)(1−( 〈D〉rodL

)2/3)2.

(4)As already mentioned, in the case of very anisomet-

ric particles, the excluded volumes are much larger thanthe actual particle volumes. Then, the contribution ofinter-particle correlations to the structure factor is muchlarger than the intraparticular contribution in the q-rangeof interest [11]. Under this assumption, the co-sweepingvolumes can be assimilated to Percus-Yevick volume frac-tions. Figure 4 displays the evolutions, predicted by thismodel, of 〈φ〉CS vs. the spherical volume fraction (fig. 4(a))and vs. the ratio of the inter-particle distance to the equiv-alent diameter (fig. 4(b)), together with the values of〈φ〉PY used in the fits of fig. 3. It must be pointed out thatthe fitted values lie beyond the range of validity of Percus-Yevick approximation, i.e. 〈φ〉PY < 0.45. The agreementbetween adjusted and calculated values is neverthelessvery satisfactory for sphere volume fractions lower than 10.A discrepancy is sometimes observed for the highestvolume fractions investigated, in particular for imogolitesuspensions. Such discrepancies occur for strongly gelledsystems, for which the assumption of a liquid-structurefactor is rather questionable. Still, the approach developedin the present letter allows in principle to calculate the

structure factor for suspensions of anisometric particles ofknown dimensions, by combining eqs. (1a) or (1b) withthe results displayed in fig. 4, provided that the systemremains ergodic.Rather simple geometrical approximations then seem to

capture the main features of the organization of suspen-sions of very anisometric particles in a wide volume frac-tion range. This may appear quite surprising becauseall investigated systems display phase transitions fromisotropic to nematic liquids and from liquids to gels. Suchtransitions, however, do not strongly alter the experimen-tal structure factors. As far as anisotropy is concerned,exploring systems of moderately anisometric particleswould certainly be relevant. The model developed heremay not describe such systems so well although thisclearly deserves further experimental investigation. Obvi-ously, the model only applies to repulsive particles as thecalculation of the structure factor involves effective hard-sphere potentials between ghost ellipsoidal particles. Finerdetails of the interaction potential, such as the electrosta-tic contribution, are not taken into account. However, inthe case of clay suspensions, ionic strength was shown toaffect inter-particle distances only slightly [7,14]. Further-more, at very low ionic strengths, i.e. when the Debyelength is of the order of or larger than the average parti-cle separation, the use of a liquid-structure factor isnot appropriate. Renormalization procedures have beenproposed to take this effect into account for sphericalparticles [21] and could in principle also be developed forlarge aspect ratios [22]. Finally, it must be emphasizedthat reaching a proper structural description of the systemis necessary to fully understand the influence of externalfields, particularly hydrodynamic ones, on suspensions ofanisometric particles.

∗ ∗ ∗

Financial support from the French ANR (ANR-07-BLANC-0194) is gratefully acknowledged. We thank DrChristian Moyne for helpful discussions and Dr DoruConstantin for his help with imogolite structure factors.

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Structure factor of very anisometric particle suspensions

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