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When depletion goes critical

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2011 J. Phys.: Condens. Matter 23 194114

(http://iopscience.iop.org/0953-8984/23/19/194114)

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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 23 (2011) 194114 (15pp) doi:10.1088/0953-8984/23/19/194114

When depletion goes criticalRoberto Piazza1, Stefano Buzzaccaro1, Alberto Parola2 andJader Colombo2

1 Department of Chemistry (CMIC), Politecnico di Milano, via Ponzio 34/3, 20133 Milano,Italy2 Department of Physics and Mathematics, Universita dell’Insubria, Via Valleggio 11,22100 Como, Italy

E-mail: roberto.piazza@polimi.it

Received 22 October 2010, in final form 22 November 2010Published 27 April 2011Online at stacks.iop.org/JPhysCM/23/194114

AbstractDepletion interactions in correlated fluids are investigated both theoretically andexperimentally. A formally exact derivation of a general expression for depletion interactions ispresented and then specialized to the case of critical correlations in the depletant by employinga long wavelength approximate analysis. A scaling expression is obtained in the critical region,suggesting a close connection to the critical Casimir effect. As a result we are able to computethe full scaling function of the critical Casimir effect in terms of the known scaling form of thedepletant equation of state. These predictions are experimentally tested in a colloidalsuspension with a micellar solution as depletion agent. Colloids are seen to aggregate reversiblywhen the micellar concentration exceeds a temperature dependent value which becomesremarkably small as the temperature approaches the lower consolution point of the micellarsuspension. Continuity between the standard depletion picture at low temperature and theCasimir effect in the critical region is demonstrated by identifying several approximate scalinglaws which compare favorably with the theoretical analysis. The transition line is seen to lieclose to the curve of maximum susceptibility of the depletant. A model, analyzed within meanfield approximation, is shown to reproduce the main qualitative features of the phenomenon.

(Some figures in this article are in colour only in the electronic version)

1. Introduction and motivation

This paper deals with the depletion forces induced in aqueouscolloidal suspensions by the addition of surfactants, andspecifically on the strong enhancements on these forces due tothe presence of a critical point in the surfactant solution phasediagram. To motivate our work, let us start with a short story.About 15 years ago, the first author of this paper was busyinvestigating the properties of fluorinated colloidal particlesthat displayed very peculiar scattering properties [1]. Thoseparticles cannot be easily synthesized in academic labs but,luckily, we had found a generous supplier in Ausimont, anItalian company that later merged into the Solvay group asSolvay-Solexis. What was really astonishing in the samplesthey provided us with was the particle concentration, which,in terms of volume fraction, was typically about 30%, a valuethat surely could not be reached in standard colloid synthesis.In fact, we were told that these concentrated samples werethe result of a ‘simple’ (so they said) post-process. A small

amount of nonionic surfactant is added to the suspension,which is then heated up close to the surfactant cloud point(i.e., close to the region where the surfactant/water mixtureundergoes a liquid–liquid phase separation). This leads to fastsettling of a concentrated colloidal phase, which is not howevera flocculated suspension, but rather very stable dispersion,free from particle aggregates. Tons of colloidal latices madeof beautifully monodisperse spherical particles are still madeevery day using such an industrial process.

By investigating the phase diagram of the system,and comparing it to what had already been found forcolloid/polymer mixtures, we soon realized that depletionforces were at work, and that colloid phase separation couldbe obtained even far from the surfactant/water miscibility gap,provided that sufficient depletant was added [2]. However,temperature effects were a deep mystery. Why does the amountof surfactant needed to induce depletion phase separationbecome so small, five to ten times smaller than what is neededat room temperature, by just raising T by a few tens of

0953-8984/11/194114+15$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

degrees? Back then, we actually guessed that there was a strictconnection to the colloid phase separation processes in criticalliquid mixtures that Beysens and Esteve had already observeda decade earlier [3]. We even found nice scaling propertiesof the phase separation line, suggesting that, as we wrote ina preliminary report [4], the increase of the correlation lengthof the surfactant solution could be regarded as ‘an increase ofthe effective micellar size, yielding a larger depletion volumewhich could more than counteract the decrease of the micellarosmotic pressure’.

However, the time was probably not yet ripe for givingthese ideas the solid basis required to arouse the interest ofa wider audience. This is the main aim of the present workthat, following and extending a recent paper [5], aims tounravel and quantitatively account for the effects on depletionforces of a demixing line in the depletant/solvent phasediagram. As we shall try to show, our experimental result andtheoretical interpretation strongly suggest that a deep relationexists between depletion forces and the critical Casimir effectoriginally predicted by Fisher and De Gennes [6], which a lucidinvestigation by Hertlein et al [7] has recently proved to bethe driving force of colloid reversible aggregation close to thecritical point of a liquid mixture. In fact, depletion mergescontinuously into the critical Casimir effect, fully sharingits scaling properties and allowing one to speak of ‘criticaldepletion’.

By anticipating the basic strategy we have followed,we wish to point out right from the start an importantdistinction that is crucial in following this paper. We planto investigate the effects on depletion forces of the presenceof a depletant critical point, namely, of a liquid–liquid (L–L) phase separation region in the phase diagram of thesurfactant solution used as a depletant solvent. However, aL–L coexistence gap is also present in the phase diagram ofa colloid where short range attractive interactions are switchedon by a macromolecular or self-associating surfactant additive.The way we shall monitor the strengthening of these attractiveforces on approaching the depletant critical point will be basedon measuring, as a function of temperature, the amount ofsurfactant required to induced L–L phase separation of thesuspension. Hence, the colloid L–L phase separation will justbe instrumental to quantify depletion effects, and should byno means be confused with the L–L phase separation of thesolvent, whose proximity is the physical cause of the observedeffects.

The paper is organized as follows. In section 2 we presenta general theoretical model, based on density functional theory,of the effect on depletion forces of strong correlation in thedepletant. The experimental system and methods are describedin section 3, whereas a detailed experimental investigation ofthe colligative and critical properties of the surfactant systemused as the depletant is presented in section 4. The mainexperimental finding discussed in [5], together with new dataconcerning the effects of additional repulsive forces on criticaldepletion, are illustrated in section 5. Finally, in section 6 wepresent a model embodying the key physical features of the realsystem, and discuss the choice of parameters able to describethe experimental colloid phase diagram.

2. Depletion effects in correlated fluids

Depletion phenomena encompass a whole class of short range,attractive, solvent-mediated interactions that take place incolloidal suspensions and are of great interest for the physicalunderstanding of aggregation phenomena in these systemsthat are relevant for technological applications. The firstdescription of the effect is generally credited to Asakura andOosawa, who recognized that when two bodies are placed in amacromolecular solution at a surface-to-surface distance suchthat macromolecules cannot be placed in between, the two arepushed together by an imbalance in the osmotic pressure ofthe solution [8]. By approximating the macromolecules asan ideal gas, they were able to derive an exact expression forthe force, the celebrated Asakura–Oosawa formula, which isrightly considered a paradigmatic description of the depletioninteraction. However, in real systems, the depletion agent, thatis the fluid hosting the colloidal particles, is often far frombeing an ideal gas and in those cases the Asakura–Oosawaformula is a crude approximation. Indeed, in a correlatedfluid the picture involving a simple exclusion of the fluidmacromolecules is not applicable; instead, the effect broughtin by two facing colloidal particles must be described interms of a perturbation in the density profile of the fluid. Anon-ideal depletant may entail a marked departure from thereference Asakura–Oosawa interparticle potential, concerningboth the shape and range: this is evident, among other factors,in studies of the depletion interaction resulting from hard-core [9] and charged [10] depletants. A subject apart isdepletion in semidiluted polymer solutions, where picturingthe depletant as a globular random coil is of course no longerpossible. In these conditions, the crucial role in setting thestrength and range of the depletion interactions is taken upby the mesh size ξ of the polymer network (for a nice andcomprehensive review, where the so-called ‘protein limit’ ofdepletion interactions is also extensively discussed, see [11]).We shall see that this problem bears strict analogies withcritical depletion.

In this section we consider a dilute colloidal suspensionin the presence of a generic correlated depletion agent and wederive a formally exact expression for the effective interactionbetween a pair of colloidal particles, valid even in the criticalregion of the depletant. Colloids are represented as hardspheres of radius R. As long as the correlation length ξ of thehost fluid is smaller than R, the effective interaction betweencolloids can be evaluated in the Derjaguin approximation,identifying the two colloids, placed at a surface-to-surfacedistance h, with two planes. Therefore, we look for theeffective interaction between two parallel surfaces placed at adistance h inside a fluid. The interaction w(z) between a planeand a molecule in the fluid depends only on the wall–moleculedistance z. To simplify the calculation we will assume thatw(z) is short ranged and vanishes beyond a distance h0. Wewill consider the regime defined by h > 2h0.

2.1. Density functional approach

We will employ density functional theory (DFT) [12] to obtaina formally exact expression for the effective interaction. If

2

J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

we denote by A[n(r)] the Helmholtz free energy of the fluidat fixed density profile n(r), we define the grand free energyfunctional �[n(r)] as:

�[n(r)] = A[n(r)] +∫

dr {�(r) − μ} n(r) (1)

by including the bulk fluid chemical potential μ and the(external) wall–fluid interaction �(r) = w(z − h

2 )+w(z + h2 ).

Here, and in the following, the origin of the coordinates is setat a point equidistant from the two planes. According to theHohenberg–Kohn theorem [12], the equilibrium profile n(r) isdetermined by the extremum condition:

δ�[n(r)]δn(r)

= 0. (2)

For symmetry reasons, in such a simple geometry, theequilibrium solution n(r)

• only depends on the coordinate z• is even in z.

Therefore the problem is effectively one-dimensional, and itis natural to introduce the free energy functionals for the unitsurface a[n(z)] = A[n(z)]/S and ω[n(z)] = �[n(z)]/S.

Let us denote the equilibrium profile that minimizes thefree energy functional ω[n(z)] by n(z). The force for a unitsurface, in the z direction, acting on the right wall is formallygiven by

F(h) = −dω [n(z)]dh

(3)

and can be more explicitly written in the equivalent forms:

F(h) =∫ ∞

0dz n(z) w′

(z − h

2

)(4)

= −∫ ∞

0dz n′(z)w

(z − h

2

)(5)

=∫ ∞

0dz n′(z)

[δa[n(z)]δn(z)

∣∣∣∣∣n(z)=n(z)

− μ

](6)

= pN (z = 0) − pbulk, (7)

where primes represent differentiation with respect to z.The simple and intuitive expression (7) is exact in

correlated fluids, independently of the thermodynamic state.It shows that the force per unit surface acting on a wall justdepends on the difference between the (normal) pressure atthe midpoint between the two walls and its bulk value. In thelimiting case of hard walls in an ideal gas it reproduces theknown Asakura–Oosawa result, but equation (7) or (6) can befruitfully applied to more general cases.

2.2. Depletion at criticality: long wavelength analysis

On the basis of the previous expressions, we now evaluatethe effective force in the case of a critical fluid. The mostconvenient starting point is equation (6), which does notcontain the wall–particle interaction explicitly, but requiresthe knowledge of the equilibrium density profile n(z). Todiscuss the long range tails of the effective interaction,we adopt an explicit expression for the density functional

a[n(z)] appropriate for describing long wavelength densityfluctuations:

a[n(z)] =∫

dz

[b

2

∣∣∣∣dn(z)

dz

∣∣∣∣2

+ f (n(z))

], (8)

where f (n) is the Helmholtz free energy density of thehomogeneous fluid. This form corresponds to a local densityapproximation plus gradient corrections, where the parameterb measures the stiffness of the fluid with respect to densityfluctuations. Clearly this functional form is inadequate todescribe the strong oscillations in the density profile of aliquid close to a wall but is meant to reproduce the overallmonotonic decay of the density profile at distances larger thanthe molecular diameter [13]. A physical insight on the structureof this density functional can be obtained by recalling that thedirect correlation function of a homogeneous fluid is definedas the second functional derivative of the excess free energy(divided by −kBT ) −β Aex with respect to n(r):

c(r − r′) = − δ2β Aex

δn(r)δn(r′)

∣∣∣∣n0

. (9)

By use of the functional form (8) we obtain, in Fourier space,

c(q) = −β[ f ′′ex(n0) + b q2]. (10)

The stiffness b is then expressed in terms of the second momentof the direct correlation function of the bulk fluid, which isknown to be of the order of the second moment of the pairpotential. We finally observe that the structure (8) is consistentwith the long wavelength limit of accurate density functionalscommonly used in the literature [14].

When equation (8) is substituted into equation (6) theforce per unit surface acting on the plane placed at z = h/2can be written as

F(h) = [ f (n0) − f (n(0))] − ∂ f (n0)

∂n0[n0 − n(0)] (11)

where the chemical potential has been expressed in terms ofthe bulk free energy density by use of standard thermodynamicrelations. The effective force is then expressed solely in termsof the free energy density of the homogeneous host fluid f (n0)

and the value of the equilibrium density profile at the midpointbetween the two walls n(0). In the case of hard walls, theformal solution of the extremum condition on the free energyfunctional provides an implicit relation between the force perunit surface F(h) and n(0), in terms of the ‘contact’ value ofthe density3 ns = n(h/2):

∫ n(0)

ns

dφ√F(h) − f (n0) − μ (φ − n0) + f (φ)

= h√2b

.

(12)Equations (11) and (12) together allow one to express F(h) interms of the boundary condition ns, which implicitly containsthe information on the wall–particle interaction.

3 Due to the coarse grained expression of the density functional (8), thecontact value is effectively the extrapolation of the long distance monotonicdecay of the density profile, rather than the true density at the wall. ns

embodies all the information on the wall–particle interaction w(z).

3

J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 1. Scaling function θ(x; z) for x = 0 compared to differentMonte Carlo simulations from [16].

In the critical region, the free energy density of ahomogeneous fluid f (n0) at a given temperature T can bewritten in terms of a universal scaling function �(x):

f (nc + δn0) − f (nc)

kBT= a11 ε2−α �(b1 δn0ε

−β), (13)

where the notation of Pelissetto and Vicari [15] has beenused. The reduced temperature is ε = (T − Tc)/Tc, δn0 =n0 − nc, while a11 and b1 are metric factors, i.e. non-universal,dimensional constants. In the following we will define thescaling variable as

x = b1 δn0ε−β. (14)

Moreover we assume that the stiffness b is finite at the criticalpoint. This is actually an approximation because, accordingto equation (10), b is known to diverge at criticality as longas the critical exponent η is different from zero4. In threedimensions η ∼ 0.04 and we can safely disregard such a weaksingularity. Therefore, in the following we explicitly set η = 0.It is convenient to introduce the rescaled quantities σ and y,together with φ0 = b1 [n(0) − n0] ε−β :

σ(x; y) ≡ − F(h)

a11kBTε−(2−α) = �(x + φ0) − �(x)

− φ0 � ′(x), (15)

y ≡√

b21a11kBT

2bhεν

=∫ φ0

−∞dφ√−σ + �(x + φ) − �(x) − φ� ′(x)

, (16)

where use has been made of equations (11) and (12) and therescaled density at contact b1 (n0 − ns) ε−β has been pushedto infinity in the ε → 0 limit. In terms of these dimensionlessand universal quantities, the effective force per unit surface inslab geometry becomes:

F(h) = −kBT

hd[yd σ(x; y)]a11

(2b

b21a11kBT

)d/2

, (17)

4 The critical exponent η governs the long wavelength limit of the structurefactor at the critical point: S(k) ∼ k−(2−η).

Figure 2. Absolute value of the scaling function θ(x; y1/ν) plotted inthe x, y plane as a blue shaded map (gray shaded in print version).The function is negative over the whole plane. The black curvesdefine thermodynamic paths at constant h and δn0.

where d = 3 is the space dimensionality. Note that theratio Fhd/kBT is dimensionless and universal, because thespecial combination of non-universal metric factors appearingin equation (17) is proportional to a universal amplituderatio [15]:

a11

(2b

b21a11kBT

)d/2

= 2d/2

g+4

(18)

with g+4 ∼ 23.6 in d = 3 leading to

F(h) = kBT

hdθ(x; y1/ν) = −kBT

hd[yd σ(x; y)]2d/2

g+4

(19)

y ≡√

b21a11kBT

2bh εν, (20)

where we introduced the universal scaling function θ(x; z)governing the critical Casimir effect. This result has beenobtained by noting that the constant b coincides with theratio between the square of the amplitude f + governingthe divergence of the correlation length, ξ ∼ f +ε−ν ,and the amplitude C+ characterizing the divergence of thesusceptibility, χ ∼ C+ε−γ . In the critical region the generalexpressions (6) or equivalently (7) reduce to a universal formstrongly resembling those governing the critical Casimir effect.This shows that the same mechanism which gives rise tostandard depletion effects in non-critical fluids is responsiblefor the long range Casimir force when critical fluctuationsset in. Note that the approximate expression (19) we havederived holds in the whole critical region, both along thecritical isochore and at non-critical density. A plot of thescaling function on the critical isochore (x = 0), comparedwith available Monte Carlo data [16] is shown in figure 1, whilefigure 2 shows the behavior of the function corresponding to arange of values of the scaling variable x . The resulting force isalways attractive and, at fixed reduced temperature ε, it decaysexponentially at large h on a characteristic length which scalesas the correlation length ξ .

In the limit of small h, on the contrary, the forceis formally divergent. This is clearly an artifact due to

4

J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

the extrapolation of the correct, long-range behavior to aregime where small-wavelength density fluctuations play amajor role that cannot be accounted for by the approximatedfunctional (8). As a consequence, the universal form (19)holds only beyond a certain cut-off distance hc. In light ofthe assumption underlying the derivation of all the previousexpressions, it is natural to assume as the cut-off distance twicethe range of the fluid–wall interaction, hc = 2h0. In theinterval 0 < h < hc a non-universal, finite force dependingon the details of the fluid–wall interaction is expected.

By looking at figure 2 one can follow the amplitude of thescaling function θ(x; y1/ν) along different paths in the (x, y)

plane. If one fixes the reduced temperature ε and the bulkreduced density δn0, which amounts to keeping x constant,the corresponding vertical path describes θ as a function ofthe distance h between the planes. If, on the contrary, h andδn0 are kept constant and the temperature is varied, the familyof black curves shown in the figure is traced out. In eithercase, it is apparent that the scaling function is quite smallalong all the paths corresponding to δn0 < 0, whereas alongthe paths corresponding to δn0 > 0 it can attain significantvalues. Therefore, on general grounds, we expect that x ∼ 0should mark the transition line between a region where Casimirforces are negligible (x < 0) and a regime where they becomerelevant (x > 0).

In this analysis, use has been made of the asymptoticscaling function (13) embodying the leading singular behaviorof the thermodynamic quantities in the critical region. Theprecise definition of the scaling fields (δn, ε) in fluids ishowever rather subtle [17] and involves field mixing: while inthe nearest neighbor lattice gas model δn and ε have the usualmeaning of reduced density and temperature respectively; in afluid the order parameter δn is generally a linear combinationof reduced density and temperature. This can be directlychecked at mean field level, when the Helmholtz free energydensity f is an analytic function of its thermodynamicvariables (n, T ). We first define a temperature dependentreference density n(T ) by the requirement

(∂3 f

∂n3

)T

= 0. (21)

Note that this condition defines the locus of maxima of theisothermal susceptibility:

χ = ∂n

∂μ=

(∂2 f

∂n2

)−1

T

. (22)

Then we expand f (n, T ) in powers of δn ≡ n − n(T ) at fixedT :

f (n, T ) = f (n(T ), T ) + c1(T )δn + 12 c2(T )δn2

+ 1

4! c4(T )δn4 + · · · (23)

where the coefficients are defined by

ck(T ) =(

∂k f

∂nk

)T

∣∣∣∣n=n(T )

. (24)

The spinodal line is defined by the divergence of the isothermalsusceptibility (22) and the critical temperature is determinedby the additional extremum condition (21). As a consequence,c2(T ) vanishes at Tc and, to leading order, can be taken as alinear measure of the reduced temperature, while c4(T ) attainsa finite limit at the critical temperature. It is then natural todefine a temperature scaling field as

ε = c2(T )

c4(T ). (25)

By use of this definition, the free energy (23) can be expressedin the standard mean field scaling form [15]

f (n, T ) = freg(n, T ) + c4(T )ε2�(x), (26)

where x = δnε−1/2 and �(x) = x2

2 + x4

4! . Note, however, thatin this way the ‘density variable’ δn = n − n(T ) involves botha change in density and in temperature.

In a lattice gas model, where particle–hole symmetryholds, the reference density n(T ) does not depend ontemperature and, as a consequence, δn just measures deviationsfrom the critical density. However, in a fluid the ‘symmetry’line where the susceptibility is maximum generally differsfrom the critical isochore, although both paths clearly end atthe critical point. The difference between the two approachesto the critical point becomes particularly clear in systemswhere the coexistence curve is noticeably skewed, leading tothe appearance of a further correction to the scaling terms,related to the presence of ‘odd operators’ in renormalizationgroup language, whose amplitudes are reduced if the specialpath δn = 0 is chosen. Field mixing is usually discussed inthe grand canonical statistical ensemble, where suitable linearcombinations of temperature, chemical potential and pressureare identified as scaling fields [18]. By carefully examiningthe behavior of the maximum susceptibility line n(T ), a weakdeviation from linearity, which closely parallels the violationof the rectilinear diameter law in the coexistence region, isfound: n(T ) − nc ∼ Bdε

2β + Adε1−α + A1ε. At mean field

level this result is consistent with a finite slope of the maximumsusceptibility line at the critical point.

As a final remark, we note that, along this line, definedby x = 0, the product between the singular contribution to thepressure �Psing = Psing − Psing(Tc, nc), divided by kBT , andthe correlation volume is constant, and universal in the criticalregion:

�Psingξ3

kBT= Q+

α(1 − α)(2 − α)∼ 0.1, (27)

where α = 0.11 is the critical exponent for the specific heatand Q+ = 0.019 is a universal amplitude ratio [15]. Moreover,along this line, due to field mixing, the correlation lengthshould diverge as ξ ∼ |n0 − nc|−ν , with a critical exponent ν

ranging from the mean field value ν = 0.5, appropriate in thequasi-critical region, to the asymptotic result ν = 0.63 [15],very close to the critical point.

5

J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

3. Materials and experimental methods

3.1. Colloidal system

The colloids we have used are aqueous suspensions ofHyflon™ MFA, a copolymer of tetrafluoroethylene (TFE) andperfluoromethylvinylether (PF-MVE), produced by Solvay-Solexis S.p.A. (Bollate, Italy), and made of spherical,monodisperse particles with an average radius R ∼ 90 nmand a polydispersity of about 4%, whose distinctive opticalproperties (in particular, their low refractive index n � 1.352,allowing one to study concentrate suspensions with negligiblemultiple scattering effects, and their partial crystallinity,yielding a depolarized scattering contribution) have beenextensively investigated in the past [1]. The surface of MFAlatex particles bears a negative charge, mostly due to thepresence of trapped fluorinated surfactant used in the emulsionpolymerization. To screen electrostatic effects, all experimentshave been performed in the presence of about 250 mM NaCl,after the particles have been sterically stabilized with a layer ofthe same surfactant used as a depletion agent (see section 3.2).

3.2. Depletant

As depletion agent, we have used C12E8 (octaethylene glycolmonododecyl ether), a nonionic surfactant, with molecularweight Mw = 538.75 g mol−1 and a density very close to1 g cm−3, and belonging to the class of ethoxylate alcoholsCmEn , where m is the number of carbon atoms in thehydrophobic chain, and n is the number of ethoxylate groupsconstituting the hydrophilic head group. Beyond its criticalmicellar concentration cmc = 0.038 g l−1 (�71 μM, [19]),at room temperature C12E8 forms globular micelles with aradius a = 3.4 nm and an aggregation number N � 95–100,as determined by both light [19] and neutron [20] scattering.The refractive index increment of C12E8 solutions in water(dn/dc = 0.134 − 2.4 × 10−4[T (◦C) − 25] l g−1 [19]) isvery weakly dependent on temperature. In our system, C12E8

also plays the role of a steric stabilizer for the particles. Whenadded to MFA dispersions, the surfactant initially adsorbs onthe particle surface, forming a compact monolayer with athickness of about 2 nm that stabilizes the suspension up tovery high ionic strength. The sample batch we have used,synthesized by ENI S.p.A., S. Donato Milanese (Italy), wasthoroughly purified by recrystallization to obtain high-purityC12E8, essentially free from residual long-chain alcohols thatcould noticeably influence its aggregation and critical behavior.

Aqueous solutions of C12E8 display an inverted miscibilitygap, bounded below by a critical demixing point. In thepresence of 250 mM, the critical temperature is Tc � 64.5 ◦C.Liquid–liquid phase separation with increasing T is not auncommon feature, for it is also shared by many simplemixtures such as 2,6-lutidine + water5. Compared to the latter,5 In fact, an inverted coexistence gap can be expected anytime the cross-interaction between the two components of a mixture become less favorableat higher T , for instance because of weakening of the hydrogen bonds. Thisinverted coexistence curve is actually the lower part of a closed-loop gap,which always ends in an upper critical point because the contribution of theentropy of mixing to the free energy becomes dominant in any case at high T .Usually, however, this additional critical point cannot be observed, because itoccurs for T larger than the boiling point of the mixture.

however, two aspects of the phase behavior of C12E8 (sharedby most ethoxylated surfactants) are rather peculiar. First, thetemperature region where critical effects can be detected (thecritical regime) is very wide, generally ranging in the tens ofdegrees, which is more than an order of magnitude larger thanfor simple binary mixtures. Second, the critical concentrationis very small, of the order of a 1–2% in weight fraction, sothat the demixing region is very asymmetrical. Whereas thefirst feature can be reasonably expected in a binary mixturewhere the size of the micelles is much larger than the solventmolecules, the second one is harder to account for using asimple model of the interaction potential6. We shall come backto these peculiar aspects in section 6.

3.3. Determination of the phase separation points

All experiments were performed in sterically stabilized MFAsuspensions at a particle volume fraction � = 0.03, in water+250 mM NaCl to which a variable amount of C12E8 was added.To determine the temperature where phase separation takesplace, the samples were sealed in glass ampules with a volumeof 1 cm3, mounted in parallel on a frame, and immersedin a water thermostat with a temperature setting accuracy±0.02 ◦C, with a window allowing for optical inspection. Thetemperature was increased in small steps (close to the transitionpoints, typically by 0.1 ◦C), and the system was allowed tothermalize for at least 10 min between each temperature step.Phase separation was detected by monitoring with a simplelaser pointer the noticeable increase of turbidity marking theonset of the phase separation processes, followed on a timescale of a few minutes by the appearance of a settling meniscus.The same method was also used to find the coexistence line(or, more precisely, the cloud point curve) of the C12E8/brinesystem in the absence of particles. We checked that the latterwas not shifted by the presence of the colloids, at least up to aparticle volume fraction of � = 0.05.

3.4. Light scattering measurements

Static (SLS) and dynamic (DLS) light scattering measure-ments, aimed to extract the physical properties of the depletantdiscussed in section 2, were performed using a custom-made setup, equipped with a frequency-doubled NdYag+laser operating at λ = 514.5 nm, and with close-circuitsample feeding for accurate dust removal. Since, as weshall discuss in what follows, the investigation range wasrestricted to conditions where the correlation length of thedepletant does not exceed about 15 nm, so that the scatteringintensity does not appreciably depend on the scatteringwavevector k, all measurements were performed at a fixedscattering angle θ = 90◦, corresponding in water to k �0.022 nm−1. This choice allowed the use of a micro-volume rectangular cell of excellent optical quality, whichis particularly useful when dealing with very dilute, weaklyscattering surfactant solutions. Viscosity measurements were

6 For this peculiar kind of phase diagram, the notion of field mixing weintroduced in section 2 is particularly relevant. Failure to appreciate thispoint is possibly at the root of the contradictory experimental results obtainedfor the critical exponents of some nonionic surfactants, which provoked aconsiderable debate in the past.

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 3. Concentration dependence of the quantity c/Is forC12E8/water solutions, in the presence of 250 mM NaCl, at fivetemperatures between 25 and 39 ◦C. Intensity data are normalized soas to have limc→0(c/Is) = 1 at T = 36.3 ◦C. The inset shows thetemperature dependence of the virial coefficient k2.

made using a calibrated Ubbelohde viscometer, inserted in thesame thermostat bath used for phase separation studies. Therefractive index of the surfactant solutions was obtained by ahigh-precision Abbemat refractometer (Anton Paar GmbH).

4. Colligative and critical properties of C12E8aqueous solutions

To test the theoretical predictions made in section 2, a carefulcharacterization of the colligative and critical properties of thedepletant, in terms of osmotic pressure and correlation length,is required. Because the Derjaguin approximation restrictsthe validity of the theoretical results to conditions where thecorrelation length of the depletant/solvent system is muchsmaller than the colloid radius, we limited the experimentalinvestigation to a ‘quasi’ critical region that does not extendtoo close to the critical point. Quantitatively, the smallestinvestigated reduced temperature is ε = (Tc − T )/Tc � 10−2,corresponding, as we shall see, to ξ � 13 nm, and therefore toξ/R � 0.14.

4.1. Low temperature: onset of attractive interactions

We first investigate the onset of effective intermicellarattractive interactions, measuring, by static light scattering, thesecond osmotic virial coefficient, as a function of temperatureT , of C12E8 in water/NaCl solutions. We recall that, forparticles much smaller than the wavelength, the scatteredintensity Is is simply proportional to the structure factor S(0)

at zero scattering wavevector, and therefore to the osmoticcompressibility of the solution:

Is = A

(dn

dc

)2

kBT c

(∂�

∂c

)−1

,

Figure 4. Concentration dependence of the scattering intensity perunit concentration Is/T for C12E8/water solutions, in the presence of250 mM NaCl, up to a temperature difference Tc − T = 4 ◦C fromthe critical value.

where kB is the Boltzmann constant and A an instrumentalcalibration constant. At low concentration, provided thatthe refractive index increment is weakly dependent ontemperature, this yields:

c

Is∝ 1

Mw(1 + k2c),

where Mw the molecular weight of the micelles, and k2 istwice the second virial coefficient of the osmotic pressure.In particular, for interactions that depend on temperature,the value of T where k2(T ) = 0 plays the role of the ϑ-temperature for a polymer solution. Figure 3, where theresults obtained in the temperature range 25 ◦C � T � 39 ◦Care plotted, shows that intermicellar interactions switch fromrepulsive to attractive at a temperature ϑ � 35 ◦C. Within thesame range, the micellar aggregation number does not vary bymore than about 10%.

4.2. Critical behavior

On further increasing T , a rapid growth of the scatteringintensity and a noticeable shrinking of the concentrationregion where the virial expansion holds take place. Figure 4shows that a pronounced peak of the scattered intensityper unit concentration Is/c develops in correspondence to aconcentration c � 1.8% by weight. With respect to the ratherfuzzy indication that can be extracted from the flat cloud pointcurve, we regard the latter as a more reliable indication ofthe critical concentration. In what follows, we shall thereforeassume cc � 18 g l−1 � 33 mM l−1. Notwithstanding the hugescattering increase around cc, Is shows a drastic drop at verylow concentration (for instance, the ratio between the intensity

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 5. Correlation length at the transition points (cs, Ts), plottedas a function of the reduced temperature (bullets, upper axis) andconcentration (squares, lower axis). Both tentative power-law fits(full lines) to the data closer to the critical point yield exponents ofabout 0.55–0.6.

scattered at T = 60.5 ◦C and T = ϑ � 35 ◦C decreasesfrom about 17 at c = cc to about 5.5 at c = 1.25 g l−1).This suggests that the scattered intensity in the vanishingconcentration limit is very weakly dependent on temperature,and that the scattering increase, in agreement with neutronscattering data, is mostly due to the critical effect, whereasmicellar growth is very limited.

4.3. Correlation length

As discussed in section 2, the correlation length ξ , the onlyrelevant length scale close to the critical point, is expectedto play a major role in setting the strength of the effectiveinteractions induced by the surfactant/water solution. Lightscattering measurements allow one to extract ξ using twoessentially independent methods, namely, (a) from the angulardependence of the static scattered intensity, and (b) from theslowing-down of the collective diffusion coefficient obtainedby DLS. The first method consists in fitting the dependenceof the scattered intensity on the wavevector q with the Fisher–Burford expression:

Is(q, ξ) = I0

(ξ−2 + q2)1−η/2, (28)

where η � 0.04 is the Fisher exponent. Equation (28)embodies the divergence of the compressibility as (Tc − T )−γ ,with γ = ν(2−η), and shows that the angular dependence of Is

is significant only provided (qξ)2 is sufficiently large. As weshall see, however, in the temperature range we investigatedξ � 13 nm, so that the maximum value (in backscattering

Figure 6. Equations of state of C12E8/water solutions in the presenceof 250 mM NaCl. Absolute calibration of � is obtaining by fixingthe slope of the straight line obtained at low concentration forT = 36.3 ◦C at the value for an ideal gas of micelles withaggregation number N = 95, as explained in the text (full line). Thesquares in fuchsia correspond to the values �(cs, Ts) shown infigure 2 of the paper.

configuration) of (qξ)2 � 0.15, which is too small to allowfor a careful determination of ξ even at the temperature valuesclosest to Tc. An alternative and effective method to obtainξ is to obtain the collective diffusion coefficient D from thedecay rate � = 2Dq2 of the intensity correlation functionof the scattered intensity. For small values of qξ , D canbe written as the first-order approximation to the generalKawasaki expression [21]:

D = hkBT

6πηsξ[1 + O(q2ξ 2)], (29)

where h � 1. This expression was used to obtain ξ forC12E8 + water + 250 mM NaCl solutions corresponding tothe phase separation points (cs, Ts) of the colloidal suspension,which is shown in figure 5.7

4.4. Osmotic pressure

The concentration dependence �(c) of the osmotic pressureof C12E8/water solutions (namely, their equation of state)for different temperatures can be obtained by numericalintegrating the compressibility data obtained by SLS. Somerepresentative data are shown in figure 6. Calibration for theabsolute values of � was obtained as follows. Since we foundthat at T = 36.3 ◦C, which is very close to the system ϑ-temperature, intermicellar interactions are extremely weak, the

7 It is important to stress that, at variance with the usual expression for theBrownian diffusion coefficient of a particle, the prefactor in equation (3)contains the dynamic viscosity ηs of the solution (not of the solvent), whichvaries considerably across the different phase separation points.

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 7. Fast sedimentation kinetics of a MFA suspension in the presence of a sufficient amount of Triton X100.

slope of the straight line giving the low-concentration behaviorof �(c) was fixed at the values predicted for an ideal gas ofC12E8 micelles with aggregation number N = 95. Namely, thepressure axis was rescaled so that, at T = 36.3 ◦C, we have:

� [Pa] = 103 RT

Nc [mol l−1], (30)

for c → 0, as indicated by the straight line in figure 6.We remark that equation (30) does represent the correctlow-concentration limit at all temperatures but, at the ϑ-temperature, the O(c2) correction vanishes, making thecalibration more reliable.

5. Critical depletion

5.1. Surfactant-induced depletion and basic principle of themeasurements

We first recall some basic features of the phase behaviorof colloidal systems in the presence of nonionic surfactantmicelles, in conditions where the latter act as an almost-ideal depletant, mainly referring to the experiments presentedin [22], where Triton X100 was used as depletion agent. TritonX100 is a nonionic surfactant bearing strict analogies withthe CmEn , so much so that all the features observed in [22]using Triton X100 are shared by the experiments presentedhere when phase separation induced by C12E8 is studied farfrom the critical point8.

8 The affinity between these very closely related surfactants extends to manyfeatures of the phase behavior. Similarly to C12E8, indeed, Triton shows aupper critical demixing point at T � 63 ◦C, cc � 2%, and the qualitativetemperature dependence of depletion effects in C12E8 solutions that arepresented in this paper are, to all intents and purposes, also shared by Triton.The latter, however, being a commercial product, is a poorly characterizedmixture of nonionic surfactants with similar chain length, and therefore notparticularly suitable for dealing with critical effects.

The experiments in [22] yield the following qualitativeobservations. By progressively increasing the amount of addedsurfactant, the colloidal suspensions switch from a behaviorwhere they remain transparent and sediment very slowly, to astate marked by sudden increase of turbidity followed by veryrapid settling, which leads in a few hours to the formation ofa dense sediment. The latter then restructures very slowly,but still stays much more expanded than the very compactsediments eventually obtained by ‘regular’ slow sedimentationat low surfactant concentration. The transition between thesetwo different regimes is abrupt, but reversible. A small increasein Triton volume fraction induces a transparent sample tobecome turbid and sediment fast. Nonetheless, if the surfactantcontent in the supernatant is slightly diluted by the addition ofa few drops of water, the sediment rapidly dissolves on gentleagitation, yielding a stable suspension. A typical sedimentationkinetics for fast-settling samples is shown in figure 7. Detailedmeasurements of the equilibrium sedimentation profiles allowone to extract an equation of state of the colloid as a functionof depletion strength which fully matches the predictions forthe osmotic pressure �(�) of a system of sticky hard spheres,and to map the experimental phase diagram onto the numericalsimulation results for depletion forces with a very short range,set by the ratio q � 0.03 between the micelle and particle size.The most noticeable feature of the phase diagram is that allrapidly settling samples are placed within, or very close to, themetastable L–L coexistence gap [23]. This strongly suggestsregarding gelation as an arrested phase separation process. Inother words, as later confirmed in a detailed study by Lu et al[24] on a different system, the spinodal decomposition processtaking place by crossing the L–L phase boundary acceleratesthe phase separation process so much that the suspension getstrapped into a frozen disordered state.

The principle underlying the present experiment istherefore the following. Reference [22] shows that fastsedimentation can be regarded as a basic clue for locating the

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

L–L phase separation boundary of a colloidal system, an ideathat was later successfully applied to investigate surfactant-induced depletion in the presence of additional long rangedelectrostatic interactions [10]. Formally, this amounts toinvoking the Noro–Frenkel extended law of correspondingstates [25], stating that all short ranged attractive potentialsare characterized by the same thermodynamic properties ifcompared at the same reduced density and dimensionless virialcoefficient of the colloid osmotic pressure:

B∗2 = 3

2R3

∫ ∞

0dr r 2(1 − e−V (r)/kBT ), (31)

where V (r) is the effective potential between two sphericalparticles of radius R. In particular, when experimentsare performed at a colloidal volume fraction � � 0.25,corresponding to the minimum of the L–L coexistence curve,the phase boundary is crossed for B∗

2 � −5.5. Since thecoexistence curve is very flat, this value does not decreaseby more than 10% even when the colloidal concentration isreduced to a few per cent in volume fraction. The points cs,giving the minimum amount of surfactant required to inducephase separation at temperature Ts, are therefore assumed tocorrespond to state points where the ‘strength’ of the effectiveattractive potential is the same. In other words, a reduction incs can be regarded as an increase of the ‘effectiveness’ of thedepletant.

5.2. Colloid phase diagram

As a preliminary measurement, 14 samples at � = 3% atsurfactant concentration 8 g l−1 � cs � 10.5 g l−1, in steps�cs = 0.025 g l−1, were placed in a refrigerated thermostatat T = 4 ◦C, and carefully remixed once thermalized.The minimum amount of C12E8 needed to induce phaseseparation (cs0 = 9.75 ± 0.25 g l−1) was found by lettingthe samples settle for about 12 h, and visually detecting thepresence or not of a concentrated colloid phase separated bya sharp meniscus from a clear supernatant. Subsequently, 24samples with 1 g l−1 � cs � cs0 were prepared, sealed inampules, and studied with the formerly described turbidimetricmethod. The two samples at lower C12E8 concentration(cs = 1, 1.5 g l−1) did not separate until the coexistencecurve of the surfactant solution, evidenced by a rapid spinodaldecomposition process with the development of a forward-scattering ring, was reached. When the latter suspensionsare brought within the C12E8/water coexistence gap by furtherincreasing T , and the surfactant solution is allowed to fullyphase separate, the particles are found to fully segregate in thewater-rich phase, witnessing that the latter preferentially ‘wets’the colloids.

Figure 8 shows that cs noticeably decreases on increasingT , approaching the value cs � cc for T → Tc. Thissensible reduction of cs on increasing T constitutes the basicevidence for the increase of depletion efficiency brought inby the correlation effect associated to the depletant criticalpoint. The inset also shows that log cs displays a lineartrend in a wide range of reduced temperature ε. A delicatepoint is however worth mentioning and commenting upon. In

Figure 8. Minimum amount of surfactant cs required to induce phaseseparation for aqueous MFA suspensions at a particle volumefraction � = 0.03 (full dots), in the presence of 250 mM NaCl. Theconsolution curve of the surfactant solution is shown by open dots.The inset shows cs/cc versus the reduced temperature ε on a semi-logscale, with a fit to the data.

principle, one may indeed wonder whether a simpler effect,such as a change in the morphology of the depletant, couldaccount for this evidence. In fact, on top of critical effects,some CmEn , particularly those with a small value of n, showan appreciable change of the micellar size as well, whichmakes it harder to separate out purely collective effects. Asdiscussed in section 5.1, this does not seem to be the case forC12E8. Nevertheless, it is important to point out that, even ifthere actually were an increase of micellar size, this could notaccount for the observed increase of depletion strength. In fact,in a simple ideal-depletant model, the strength of depletioninteraction would conversely decrease as a consequence of theincrease in the radius a of the depletant particles (at constanttotal depletant concentration). In the Asakura–Oosawa modelthe dimensionless second virial coefficient (31) is indeed anincreasing function of a at constant surfactant concentration.

5.3. Dependence of the state points on the depletant osmoticpressure and correlation length

In view of the theoretical model presented in section 2, it isuseful to compare the transition temperatures with the valuesof the osmotic pressure and of the correlation length of the(particle-free) C12E8/water solutions, obtained at the same statepoints (cs, Ts) shown in figure 8. Panel A in figure 9 shows that,sufficiently close to Tc, the difference cs − cc approximatelybehaves as a power-law cs − cc = �(cs, Ts)

n , with a fittingexponent n � 2/3.

It is worth noticing a marked discrepancy of theseresults with what one might naively expect by extendingthe ideal-gas model to an interacting depletant. In the AO

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 9. Excess surfactant concentration cs − cc with respect to the critical concentration cc to induce phase separation, versus the osmoticpressure (panel (A)) and the correlation length ξ(cs, Ts) (panel (B)) of particle-free surfactant solutions, both measured at the transition points.Full lines are power-law fits.

model the depletion potential is proportional to the osmoticpressure of the (ideal) depletant. When the depletant self-interaction is repulsive, the osmotic pressure increases, leadingto an enhancement of the colloid attraction induced bydepletion. On the other hand, however, the depletant alsodisplays structural correlations, and these structuring effectsare expected to reduce the attractive depletion forces. In thecase of electrostatic self-repulsion, where these two effects areconflicting, we recently found [10] that, at least for highlycharged depletant, the first contribution is still dominant.Conversely, in the critical system we are now investigating,the micelles display effective attractive self-interactions, andthey get more and more correlated approaching the criticalpoint. Both effects are then expected to weaken the depletionattraction. Yet, on the basis of our interpretation of the locus ofthe transition points, the results presented in figure 6 show thatthe state points obtained, for instance at T = 25 and 60.5 ◦C,must correspond to the same strength of the depletant, althoughits osmotic pressure is, in the latter case, almost two hundredtimes lower. The model in section 2 conversely shows that,as far as the strength of the effective interparticle interactionsis concerned, the osmotic pressure is overcompensated by thestrong increase in the range of the depletion potential. In otherwords, it is the correlation length ξ which takes on the roleof the only relevant length scale setting the range of depletionforces close to the depletant critical point. Indeed, figure 9(B)shows that cs − cc also scales as a power-law cs − cc ∝ξ(cs, Ts)

−m of ξ , where experimentally m � 1.8; which indeedlies within the expected theoretical range (1.59, 2) mentionedat the end of section 2.

5.4. State points and maximum susceptibility line

The theoretical model developed in section 2 predicts thatcritical depletion effects reach their maximum close to theχmax line on the phase diagram of the depletant/solvent system

Figure 10. Intensity scattered by C12E8 solutions at varioustemperatures, shown in the legend, compared to the locus of thecolloid transition points. Full lines are quadratic fits of c/I .

where the susceptibility ∂n/∂μ is maximal, which does notcoincide with the critical iso-concentration line. The formerχmax line can be found by evaluating the locus of the maximaof the scattered intensity as a function of concentration foreach investigated value of T . We can therefore expect the statepoints marking colloid phase separation to lie close to that line,because, as discussed in section 2.2, x ∼ 0 marks the transitionbetween a region where Casimir forces are negligible (x < 0)and a regime where they rapidly become relevant (x > 0). Thisprediction is fully confirmed by the data in figure 10, wherethe curves of maximum scattering and the transition line arecompared.

Finally, since according to equation (27), the productbetween the singular contribution to the pressure and thecorrelation volume is constant and universal along themaximum susceptibility line, we should expect a similar

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 11. The quantity �ξ 3/kBT versus the reduced temperature ε,together with its average value (full line) and one standard deviationbounds (dashed lines).

behavior for the quantity (�s − �c)ξ3, where � is the

osmotic pressure measured at the transition point, and �c is theosmotic pressure at the critical point of the surfactant solution(which, since the critical concentration is very low, and ∂�/∂cvanishing, is almost negligible). In fact, the two empiricalpower-law trends shown in figure 9 imply �(cs, Ts) ∝ξ(cs, Ts)

−2.7. Figure 11 indeed shows that the product �ξ 3

in almost independent of T across the whole investigatedtemperature range Tc–T = 4–40 ◦C, with an average value�ξ 3/kBT � 0.08 that is in remarkable agreement with thetheoretical predictions.

5.5. Particle size dependence and electrostatic interactionseffects

In view of the discussion in section 6, it is useful to inquireabout the dependence of the colloidal phase diagram onparticle size. Although we have not yet performed thisinvestigation on the system MFA + C12E8 (a task requiring alarge amount of this expensive surfactant), a set of data relativeto polystyrene particles, with radii R ranging between 19and 120 nm, suspended in aqueous solutions of Triton X100,which, as we have already mentioned, also show a criticalpoint (at T � 63.5, cc � 2%), has already been collected andpresented in [4]. Figure 12 shows that, once the amount ofsurfactant cs required to phase separate is rescaled to its valuecs0 far from the critical point, the transition lines approximatelycollapse on a single curve. Moreover, the inset shows that thevalues cs0 are fitted quite well by c−1

s0 = A + B R.We have also performed a preliminary investigation of the

effect on critical depletion of additional repulsive interactionsbetween the colloidal particles. Figure 13 shows that theelectrostatic interactions substantially increase the amount ofsurfactant needed to induce colloid phase separation. Thiseffect can be qualitatively understood if, as made in section 6,the transition line is interpreted as the locus of the pointswhere the second virial coefficient of the effective colloidinteraction attains a fixed value B∗

2 ∼ −6, since repulsiveinterparticle interactions would shift the line towards higher

Figure 12. Minimum amount of surfactant cs, rescaled to the valuecs0 obtained at 25 ◦C (Tc − T � 40 ◦C), for polystyrene particles witha radius R of 19 (dots), 35 (diamonds), 60 (squares), and 120(triangles) nm, dispersed in aqueous solutions of Triton X100. Theinset shows that c−1

s0 depends linearly on R (redrawn after [4]).

depletant concentration values. Moreover, the figure showsthat (cs − cc)/cc is a monotonically increasing function ofthe reduced temperature that, at variance with what we foundfor the highly screened regime, shows no saturation for largevalues of ε. In practice, this means that, in the absence ofadded salt, as much as 18% by weight of C12E8 is required toinduce phase segregation. One may therefore wonder whether,for sufficiently strong repulsive interactions, a condition maybe reached where ‘standard’ depletion is not sufficient to drivephase separation far from the critical point. In this case,the region of instability in the colloid phase diagram wouldbecome a closed ‘pocket’ to the side of the depletant criticalpoint, similarly to what is found for the critical Casimir effectin simple liquid mixtures.

6. Model system

As a practical illustration of the formalism developed insection 2 we now estimate the transition line in a model ofa colloid suspension within the mean field approximation.One of the most intriguing features of the phase diagramof the micellar suspension experimentally investigated in theprevious sections is the form and location of the coexistencecurve: the critical volume fraction is extremely small (fewper cents) and, as a consequence, the coexistence curveis considerably skewed. This class of systems has beenextensively investigated both experimentally and theoreticallyin the past, however no consensus has emerged yet onthe appropriate way to model micellar suspensions and onthe origin of such a remarkable asymmetry. Aside frominterpretations based on change of the aggregation number [26]

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

Figure 13. Comparison of the temperature dependence of thereduced surfactant concentration (cs − cc)/cc in no-salt conditions(squares) and in the presence of 250 mM NaCl (bullets).

or of the shape of the micelles [27], which, as we havealready mentioned, strongly depends of the specific kindof surfactant and are found to be very modest for C12E8,possible explanations range from a temperature dependenceof the hydration layer [28, 29] to the relevance of manybody forces [30], from the analogies to the behavior ofpolymer blends [31, 32] to the presence of short rangeinteractions [33]. These distinctive features of the phasediagram are probably relevant also in relation to conflictingmeasurements of the critical behavior of C12E8 aqueoussolutions [34, 19, 35, 36], which has provoked a noticeabledebate in the past9. The former anomalies may be related tothe difficulty of identifying the correct scaling fields for thisfamily of micellar suspensions. One of the consequences is thatthe critical iso-concentration line (what would be the criticalisochore for a one-component fluid) does not necessarilycoincide with the maximum susceptibility line, which arguablycorresponds to the most convenient path to approach thecritical point.

Here we will not address the open problem of physicallysensible modeling of a micellar solution, but we rather focusour attention on the consequences of the presence of a lowvolume fraction critical point of the depletant on the transitionline of a colloidal suspension. We therefore choose to analyze

9 In a series of studies of C12E8 critical solutions, Corti and Degior-gio [34, 19, 35] reported critical exponents for the osmotic compressibility andthe correlation length that were much smaller than those expected for the 3DIsing universality class (actually slightly lower than than mean field values).These apparent anomalies were analyzed in terms of corrections to the scalingby Fisher [6], and of background effects by Rouch et al [37]. Later, however,Dietler and Cannell [36] reported conflicting evidence, conversely suggestingfully universal exponents for the same system. Yet, in their closing remarks,the same authors mention that, performing careful measurements on the samplebatch used in [34], they had obtained results identical to those of Corti andDegiorgio. This comment surely calls for further investigation of the criticalbehavior of nonionic surfactant solutions.

at the mean field level a simple model, unrelated to the systemwe have experimentally investigated, displaying aggregation atlow volume fraction: a patchy sphere fluid.

The free energy of the host fluid is given by the Wertheimmodel for an associating fluid of interacting spheres ofdiameter σ , characterized by a small functionality (three activesites per molecule, see for instance [38]):

f (n)

kBT= fHS(n)

kBT+ n ln(1 − p)3 + 3

2np (32)

p

(1 − p)2= 3nσ 3eu/kB T (33)

where p is the bond probability, fHS(n) is the free energydensity of a hard sphere fluid, here represented by theCarnahan–Starling expression [39], while u is an energy scalegoverning the single site–site interaction. This model is knownto display a remarkably skewed coexistence curve with a lowdensity critical point [38], somehow similarly to the micellarsuspension experimentally investigated.

By inserting this expression into equations (11) and (12)we obtain the force per unit surface F(h) as a function of thedensity and temperature. The density at ‘contact’ ns appearingin equation (12) has been set to zero, while the ‘stiffness’parameter b has been fixed to b = 0.35uσ 5. In a microscopicmodel, like the one we are investigating, the small h cut-offintroduced in section 2 naturally emerges. In fact, whenever thedistance between the two planes is smaller than the hard spherediameter of the depletant, the density profile of the ‘fluid’ inbetween the two walls vanishes identically. As a consequence,n(0) = 0 and equation (11) gives the usual Asakura–Oosawaexpression, which is clearly regular in the h → 0 limit. Afterevaluating the force per unit surface F(h) in the whole phasediagram of the host fluid within the Derjaguin approximationvia equations (11) and (12), it is straightforward to calculatethe colloid pair potential v(h):

v(h) = π R∫ ∞

hdh′(h′ − h)F(h′). (34)

The upper limit of integration, formally identified as +∞,actually corresponds to a distance of the order of the colloidradius. However, as long as ξ < R, the contribution ath′ � R is negligible and the effective interaction is, as usual,attractive and short ranged. The dependence on the colloidradius is entirely contained within the proportionality factor,making the effective attraction stronger the larger is the colloid.Equation (34) allows one to write the formal expression forthe dimensionless second virial coefficient of the colloids B∗

2 ,which is known to play a key role in the determination of thephase diagram via the Noro–Frenkel correspondence law [25]:

B∗2 = 4 − 6R−1

∫ ∞

0dh [e−πβ R

∫ ∞h dh′(h′−h)F(h′) − 1]. (35)

Colloid aggregation is expected when B∗2 is sufficiently large

and negative: B∗2 ∼ −6. The transition lines corresponding to

different ratios between the diameter of the colloid (2R) andthat of a depletant molecule σ are shown in figure 14, togetherwith the locus of maxima of the isothermal susceptibility ∂n

∂μ

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Figure 14. Mean field phase diagram of the associating fluid model: spinodal line (dashed–dotted curve at the top left) and locus ofcompressibility maxima (full line). Left panel: transition lines of a colloidal suspension with such a depletant (dashed lines) for three ratios ofthe colloid/depletant diameter λ = 5, 10, 20. Right panel: same lines, where the concentration φ has been rescaled by a system dependentconstant φ0 in such a way that all the curves pass through a common point (λ = 5, φ0 = 1.278;λ = 10, φ0 = 1; λ = 20, φ0 = 0.733).

in the host fluid, showing a remarkable similarity with theexperimental data discussed in section 5. In particular, we seethat the smaller is the size of the colloidal particle, the larger isthe volume fraction of depletant required to induce aggregationin the suspension. Moreover, the transition line always showsthe same qualitative behavior, bending towards the criticalpoint when the critical temperature is approached. By asuitable rescaling of the volume fraction φ, the transition linescan be made to collapse, at least far from the critical region: theinverse reference volume fraction φ0 is approximately linear inthe colloid radius, similarly to what is experimentally observedin the inset of figure 12.

7. Conclusions

Summarizing, we have shown that depletion interactions andfluctuation induced forces near a critical point have a commonphysical origin. This statement is strongly supported by thecontinuity of the colloid transition line from a depletion regimeto a critical Casimir domain, the constancy of the productof the pressure times the correlation volume of the depletantalong the same line, and the strict relation of the latter tothe locus of the susceptibility maxima. The general modelwe have developed can be regarded as a first quantitativeapproach to colloid behavior in strongly correlated fluids, andmay also prove useful for scrutinizing, and possibly exploiting,other challenging effects in disperse and biological systems.We finally point out that the scaling relations we found forcritical depletion bear many resemblances to what is foundfor depletion effects in semidilute polymer solutions [11](of course, this is not too surprising, because of the formalanalogy between critical systems and polymer solutions). Inparticular, notice that the scaling law (27) for the osmoticpressure along the maximum susceptibility line coincides withthe usual scaling of the mesh size, playing the role of thecorrelation length in polymer networks. Exploring this analogy

further may yield an organic picture of the effective colloidalinteractions induced by correlated depletants.

Acknowledgments

Since this is a special issue of JPCM aiming to celebrate HenkLekkerkerker, we first wish to acknowledge, through a personalrecollection, our guest of honor, who has been influential inpromoting our interest in the subject of this work. At the timewhen the first author of this paper became interested in colloidphase separation effects induced by surfactants, studies ofdepletion effects were still in their infancy; or perhaps in theiryouth, for Henk, together with other distinguished authors, hadalready published two years before a paper that was destinedto become a landmark in colloid science [40]. AlthoughBibette and coworkers had already shown that depletion effectsbrought in by excess ionic surfactants could profitably beexploited to fractionate SDS emulsions [41], investigatingsurfactants rather than polymers as depletants was still ratheruncommon. Thus, when we submitted our first paper onsurfactant-induced depletion [2], we were quite anxious aboutthe outcome of the refereeing process. In fact, the report byone of the referees raised some criticisms, pointing out too thatthe way we had presented our arguments was rather obscure.

That referee was Henk. But Henk is a very straight person,so he soon resolved to give himself away at a meeting we bothattended, renouncing the comfortable position of anonymousreviewer for the sake of fathoming out what we had found.A full afternoon of heated, exhausting, but also illuminatingdebate was the natural consequence. Back home, we workedhard in an attempt to improve the manuscript, preparing arevised version that took into account the insights we gainedfrom discussions with Henk. Yet, this was totally wastedtime, for a few days later we received a letter from the Editorof Europhysics Letters informing us that, on the basis of thereferee reappraisal, our paper had been accepted in its current

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J. Phys.: Condens. Matter 23 (2011) 194114 R Piazza et al

form with no need for further reviewing. This is Henk. Andthis is, in our view, a splendid way to serve as a reviewer.

We also thank Solvay-Solexis (Bollate, Italy) and Eni(S Donato Milanese, Italy) for having kindly donated us theoriginal particles and depletant sample batches, C Bechinger,P Cicuta, V Degiorgio, S Dietrich, D Frenkel, and F Sciortinofor useful discussions and comments, and the Italian Ministryof Education and Research (MIUR) for PRIN 2008 funding.

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