Depletion forces in bulk and in confined domains: From Asakura–Oosawa to recent statistical physics advances

Current Opinion in Colloid & Interface Science 20 (2015) 32–38

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Current Opinion in Colloid & Interface Science

j ourna l homepage: www.e lsev ie r .com/ locate /coc is

Depletion forces in bulk and in confined domains: FromAsakura–Oosawato recent statistical physics advances

Andrij Trokhymchuk a,b,⁎, Douglas Henderson c

a Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv 79011, Ukraineb Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, Lviv 79013, Ukrainec Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA

⁎ Corresponding author.E-mail address: [email protected] (A. Trokhymchuk).

http://dx.doi.org/10.1016/j.cocis.2014.12.0041359-0294/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Article history:Received 31 October 2014Received in revised form 4 December 2014Accepted 5 December 2014Available online 16 December 2014

Keywords:Depletion interactionStructural interactionsOrnstein–Zernike equationCorrelation functionsMean spherical-Percus Yevick approximation

This article summarizes recent theoretical research concerned understanding depletion forces. These forcesappear when small colloidal particles, polymers or other entities, usually called depletants, are driven out fromthe gap orfilmbetween twomacroparticles. Applying themodern tools of statisticalmechanics and in agreementwith experimental measurements it is shown that depletion attraction is only a part of more general medium-mediated interaction forces in colloidal dispersions known as structural forces. Finally, it highlights the importantrole that long-range structural forces could play in various applications relying on effecting particle dispersion,foam and emulsion stability, as well as wetting behavior of colloidal fluids on solid.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Numerous colloid-based products, such as emulsions, foams, gels,polymer latexes, paints, inks, and coatings, widely known as colloidaldispersions, have one important common feature — their constituentsare characterized by extremely different size dimensions (see Fig. 1for visualization). These could be air bubbles or liquid droplets thatare extremely large when compared to the nanosized latex or micellarcolloidal particles as well as nanosized colloids themselves are largeenough when compared to the species of molecular solvents in whichthey are dispersed. In all these cases, the surfaces of the larger particlesserve as a spatial confinement for the suspending medium of smallerspecies, altering properties of this medium that culminates inmedium-induced (effective) interaction forces between larger particles.It is evident, that understanding the forces acting between macro-particles (or surfaces) becomes an important problem for basic as wellas for applied science, since interaction forces govern the physicalproperties of the entire system, affecting both the quality of colloidalproducts as well as the quality of processes that involve colloidal parti-cles, such as coagulation, flocculation, sedimentation and lubrication.

In the case of an athermal suspending medium, such interactionforces are essentially entropic, as was demonstrated first by Asakura

and Oosawa [1], who suggested the mechanism of effective interactionbetween two large solutes immersed in a fluid of non-adsorbingpolymers: when the separation distance between solute macrosurfacesis less than the size of suspending fluid species the latter are expelled(depleted) from the gap between the macrosurfaces leading to ananisotropy of the local pressure. The so-called Asakura–Oosawa (AO)depletion interaction is always attractive, if the separations betweenthe two macrosurfaces are less than the diameter of depletant species,and become zero otherwise.

A suspendingmedium in the AO approach ismodeled by amutuallynoninteracting (penetrable, or ideal) polymer particle that significantlyreduces the effects of the depletantfluid structure. This seems to be jus-tified for colloid–polymer mixtures where the AO depletion mecha-nism has been verified by analyzing the phase separation behaviorand flocculation of colloids with added non-adsorbing polymers aswell as by directmeasurement in a variety of colloid–polymermixtures[2,3]. However, in general, a more appropriate model for the depletantmedium is a fluid of hard-core or hard-sphere particles [4–6]. The effec-tive forces between the macrospheres dispersed in a hard-sphere sol-vent have been studied by means of modern tools of statisticalphysics such as integral equation theory [7–9], density functional theo-ry [10–14], and computer simulations [9,15–19]. The results of thesestudies have shown that taking into account the hard-core repulsionbetween particles of suspending medium emphasizes the importanceof interparticle correlations, i.e. depletant structure, and brings newfeatures to the depletant-mediated interaction. The most intriguing of

Fig. 1. Different scales of colloidal dispersions: a molecular scale (a few angstroms) of the primary suspending medium due to water, some organic solvents and electrolyte ions; a sub-microscopic scale (1 nm to 100 nm) that encompasses nanoparticles or surfactant aggregates called micelles; a microscopic scale (1 μm to 100 μm) that characterizes the size of liquiddroplets or bubbles in emulsions or foam systems; and a macroscopic scale.

33A. Trokhymchuk, D. Henderson / Current Opinion in Colloid & Interface Science 20 (2015) 32–38

these is a repulsive energy barrier, located in front of the attractive de-pletion well, and followed by the energy oscillations extended up toseveral depletant particle diameters.

Experimentally oscillatory effective interactions were found to arisewhen two macrosurfaces are immersed in supramolecular solutionssuch as colloidal and biocolloidal suspensions [20–26] and micellarsolutions [27,28] as well as have been observed also in pure molecular(aqueous and organic) solvents [29]. The latter is an important observa-tion that indicates that the interaction induced between a pair ofmacrosurfaces resolves the discrete structure of suspending mediumat length scales as small as that of the molecular solvent. Moreover,surprisingly it has been found that a hard-sphere fluid that is a rathercrude model of a molecular medium, reproduces remarkably well themain features (oscillations and exponential damping) of experimentallymeasured forces [29].

In what follows, first, we outline the Ornstein–Zernike integralequation approach within statistical mechanics and show that, indeed,the depletion interaction forces are only a part of more generalmedium-mediated interaction forces in colloidal dispersions known asthe structural forces [30]. Secondly, wewill present a simple calculationof structural anddepletion forces in bulk colloidal suspensions aswell asin colloidal films. Finally, this highlights the important role that thelong-ranged structural forces could play in various applications relyingon effecting particle dispersion, foam and emulsion stability, as well aswetting behavior of colloidal fluids on solid.

2. Integral equation approach to solvent-mediated interactions incolloidal dispersions

The microscopic properties of a discrete depletant medium nearmacrosurfaces, that are the route to solvent-mediated interactions incolloidal dispersions, can be obtainedwithin statisticalmechanics eitherby computer simulation (also called a computer experiment) or bymeans of semi-analytical approaches such as the density functionaltheory (DFT) or integral equation theory.

2.1. Ornstein–Zernike equation

An important ingredient for all theoretical approaches is theOrnstein–Zernike (OZ) equation [31]

hi j R12ð Þ ¼ ci j R12ð Þ þX3k¼1

ρk

Zhik R13ð Þck j R32ð Þdr3; ð1Þ

where subscripts i, j, and k run over all the depletant medium compo-nents 1 and 2, as well as the macroparticles G (component 3 is equiva-lent to G that is the abbreviation of giant), ρi is the number density of theparticles of component i while r1 is the position of any particle 1 andR12 = |r1 − r2| is the distance between particle 1 and particle 2.

The functionshij(R)= gij(R)− 1 are called the total correlation func-tions for a pair of particles of species i and j that are separated by the dis-tance R. The functions gij(R) are the pair or radial distribution functions.The functions cij(R) are the direct correlation functions that specify thedirect correlations between two particles. The convolution integral inright-hand-side of Eq. (1) is the indirect part of the total correlationfunction.

Many model systems consist of particles with a hard core

ui j Rð Þ ¼ ∞; R b di j0; R ≥ di j;

�ð2Þ

where dij = (di + dj)/2 and d1, d2 are the diameters of suspending fluidspecieswhile d3≡ dG=D is the diameter of the large solute. The infinityfor R b dij reflects the mutual impenetrability of the pair of interactingparticles due to the presence of a hard core. Of course, real particles donot have a hard core. However, the repulsive part of the pair potentialis steep so that approximating this repulsive part by a hard core is notunreasonable. A closure that is extremely useful for hard-core systemsis the mean spherical approximation (MSA) [32],

hi j Rð Þ ¼ −1; R b di j; ð3Þ

34 A. Trokhymchuk, D. Henderson / Current Opinion in Colloid & Interface Science 20 (2015) 32–38

together with

ci j Rð Þ ¼ −βui j Rð Þ; R ≥ di j; ð4Þ

where β= 1/kT, k is the Boltzmann constant and T is the temperature.Eq. (3) is an exact statement of the fact that particles with a hard

core cannot overlap. Eq. (4) is an approximation and can be regardedas the linearized form of the more sophisticated hypernetted-chain(HNC) approximation [33]. The advantage of the MSA is that useful an-alytic solutions can be obtained for a variety ofmodel systems, includinghard-sphere fluids. The HNC approximation does not yield analyticsolutions. The Percus–Yevick (PY) approximation is identical to theMSA for hard-sphere fluids. Because the MSA yields analytic results,our attention here will be directed to this approximation. For historicalreasons, when applied to hard-sphere systems, the MSA will be calledthe PY approximation.

2.2. Solvent-mediated interaction

Returning to the OZ equation (Eq. (1)) and letting the density ρG ofgiant particles be vanishingly small, we obtain a system of threeequations

hi j R12ð Þ ¼ ci j R12ð Þ þX2k¼1

ρk

Zhik R13ð Þck j R32ð Þdr3; ð5Þ

hiG R12ð Þ ¼ ciG R12ð Þ þX2k¼1

ρk

ZhGk R13ð Þcki R32ð Þdr3; ð6Þ

and

hGG R12ð Þ ¼ cGG R12ð Þ þX2k¼1

ρk

ZhGk R13ð ÞckG R32ð Þdr3: ð7Þ

The first equation, Eq. (5), is just the OZ equation for the homoge-neous fluid mixture. Eqs. (6) and (7) involve the giant particles and,due to this, play an important role in the statistical mechanics of con-fined (inhomogeneous)fluid systems, i.e., systems involvingmesoscopicobjects.

Eq. (6) usually yields the local density profiles ρi(R)= ρigiG(R) of thesuspending fluid species in the vicinity of a giant particle (or near asingle wall). Eq. (6) was obtained by Henderson, Abraham and Barker[34] and is commonly called the Henderson–Abraham–Barker (orHAB) equation. In addition, it yields the local density profiles ρi(z, x) be-tween the two surfaces separated by a normal distance or gap widthx= R12 − D between the surfaces of the two giant spheres (sometimesinstead of x we will use notation H that is more common to denote thethickness of a slit-like film). Statistical mechanics provides a relationbetween the local density ρi(z, H) and the force per unit area, f(H),that the species of suspending fluid exert on the inner side of the slit-like film surfaces in the direction normal to the surfaces,

f Hð Þ ¼ −kTX2i¼1

Z H

0

∂uiG z;H0� �∂z ρi z;H

0� �dz; ð8Þ

where uiG(z, H) is the “bare” fluid–surface interaction betweenfluid species i on the distance z from confining surfaces fixed at separa-tion H. Very often the force f(H) is referred to as the solvation force;f(H = ∞) corresponds to the bulk fluid pressure p. The solvation forcemeasured relative to the bulk pressure defines the so-called disjoiningpressure, Π(H) = f(H) − p. In practice, the disjoining pressure can bemeasured bydisplacing oneof the surfaces a distance dH and calculatingthe work per unit area to bring the surfaces from the separation H to

infinite separation. Thus, the film energy per unit area, W(H), can becalculated as

W Hð Þ ¼Z ∞

HΠ H0� �

dH0: ð9Þ

In turn, Eq. (7) describes the correlations between two giantparticles mediated by the suspending fluid, and usually serves to obtainthe interaction energy w(R) between a pair of giant spheres via theexact relation

w Rð Þ ¼ −kT ln gGG Rð Þ: ð10Þ

The function w(R) also is known as the potential of mean force. Inparticular, applying the HNC closure,

hHNCGG Rð Þ−cHNCGG Rð Þ ¼ βuGG Rð Þ þ ln gHNCGG Rð Þ; ð11Þ

to describe the correlations between a pair of giant spheres, the interac-tion energy due to suspending medium can be in a simple mannerexpressed via the linearized PY correlation functions

βw R12ð Þ ¼ ln gHNCGG R12ð Þ ¼ gPYGG R12ð Þ−1

¼ βuGG R12ð Þ þX2k¼1

ρk

ZhGk R13ð ÞckG R32ð Þdr3:

ð12Þ

By differentiating the interaction energy, the force F(R)=−∂w(R)/∂Rbetween two macrospheres can be easily obtained [7].

We note that the linear relation (Eq. (12)) betweenw(R) and gGGPY (R)

is not the result of the linearization of the exact expression (Eq. (10)).This is a consequence of an exact relation within the linear PY andnon-linear HNC theories that sets up a physically correct way to exploitthe PY equation for hard-sphere solutes. The utility of Eq. (12) is that theintegral in the right-hand-side of this equation can beperformed analyt-ically by using the known PY solution [35]. In this case, using the linearrelation betweenw(R) and gGG

PY (R) it is equivalent to say that the PY ap-proximation has been employed to describe fluid–fluid and fluid–solutecorrelations but in obtaining the solute–solute radial distributionfunction, gGG(R), the HNC approximation is used.

2.3. Spherical and plane geometries

Eq. (6) in conjunction with Eqs. (8)–(9), provides a description of aslit-like geometry while Eq. (7) combined with Eqs. (10)–(12) providesa route to the sphere–sphere geometry. The connection between thesetwo geometries can be obtained from the Derjaguin construction [36]

W Hð Þ ¼ − 2πD

F Rð Þ ¼ − 2πD

∂w Rð Þ∂R ; ð13Þ

which gives the energy per unit area of two flat surfaces in terms of theforce per radius between two giant spheres. We recall that here R is thecenter-to-center distance between two macrospheres and H is theseparation between two plane parallel surfaces. Another form of theDerjaguin construction concerns the disjoining pressure between twoflat walls that can be expressed through the second derivative of the po-tential of mean force between two giant spheres

Π Hð Þ ¼ −∂W Hð Þ∂H ¼ − 2

πD∂2w Rð Þ∂R2 : ð14Þ

Eqs. (13) and (14) are obtained by regarding the giant spheres asbeing composed of a collection of planar integration elements andintegrating. If we suppose that the same approximation has been usedin the set of OZ equations (Eqs. (5)–(7)), then Eqs. (13) and (14) canbe used to validate the Derjaguin construction.

Fig. 2. Oscillating interaction energyw(x) between two macrospheres of diameter D, thatare suspended in a solvent composed of small hard-sphere particles of the same diameterd1 = d2 ≡ d = 0.1D. Two cases that correspond to different values of suspending fluidvolume fraction η, are presented. The solid lines are the results of Eq. (15) while symbolsrepresent the results of parametrized equation [10] based on the asymptotic form(Eq. (18)). This parametrization provides an accurate fit to DFT results due to Roth et al.[10] as well as to computer simulation data [9]. At lower volume fractions the curvestend to reproduce the AO result [1].

35A. Trokhymchuk, D. Henderson / Current Opinion in Colloid & Interface Science 20 (2015) 32–38

3. Interaction forces mediated by hard-sphere solvent

The integral in the right-hand-side of Eq. (12) can be performed an-alytically by using the known PY solution [35] for correlation functionsof the hard-sphere mixture. Here we present this result in a form ofthe Laplace transform L[βw(R)] of the solvent-mediated interaction

L βw Rð Þ½ � ¼ −s3 η1e

−sd1 þ η2e−sd2

� �− 3

2δ2s

2−3δ1sþ 3δ0

� �

he−s d1þd2ð Þ−L1 sð Þe−sd1−L2 sð Þe−sd2 þ S sð Þ D; ð15Þ

where h=36α1α2(d1− d2)2,αi= πρi/6 and δn= π(ρ1d1n+ ρ2d2n)/6. Thecomplete derivation of Eq. (15) as well as the expressions for functionsL1(s), L2(s) and S(s) are outlined in Ref. [37].

A knowledge of the analytical expression for solute–solute interac-tion energy w(R) in Laplace space easily provides us with analyticalexpressions for all other important properties, namely, the force F(R)between solutes, energy per unit area W(H) and disjoining pressureΠ(H) between two surfaces which are related to w(R), in accordancewith Eqs. (13) and (14), by means of differentiation over distance thatcorresponds to multiplication by variable s in Laplace space. Additional-ly, the Laplace transforms consist of the information on the values of thecorresponding property at the contact distance R= D (or H= 0) usingthe relation:

w 0ð Þ ¼ kT lims→∞

sL βw Rð Þ½ �: ð16Þ

3.1. Solute spheres in a monodisperse hard-sphere solvent

A Laplace transform (Eq. (15)) may be inverted to real space usingthe usual expression involving a line integral [38].

A convenient equivalent method of inverting the Laplace transformis obtained by substituting s = ±ik into the Laplace transform to pro-duce the Fourier transform [39]. Since the Laplace transform is knownanalytically, the Fourier transform is also analytic. This Fourier trans-form is inverted by means of the usual Fourier integration [38]. Moreinformation on the details of the inversion procedure can be found inour recent studies [37].

Fig. 2 shows the interaction energy obtained from Eq. (15) togetherwith the DFT-based results of Roth et al. [10]. The DFT result for interac-tion energy has been obtained in away that satisfies several consistencyrequirements and is practically indistinguishable from computer simu-lation data. Note, that the energy is oscillatory and exhibits stratifica-tion. As the two giant spheres are brought closer together, layers ofsuspending fluid species are ‘squeezed’ out. At very close separations,all of the suspending fluid species have been squeezed out. When thegap between the pair of giant spheres is depleted of the species ofsuspending fluid one may speak of a depletion force that is attractive.This effect was first noted by Asakura and Oosawa [1].

Alternatively, the inverse Laplace transform (Eq. (15)) can be evalu-ated analytically by residues. An advantage of the inversion bymeans ofresidue theorem is that it provides one with simple analytical results atshort distances as well as in the asymptotic regime of large R. In partic-ular, for the interaction energy between a pair of giant spheres at theircontact, x = 0, we obtain

βw 0ð Þ ¼ −32

δ21−δ3ð Þ2 D; ð17Þ

while the asymptotic form at large R reads

βw Rð Þ≈−2 γj j cos ωRþ argγð Þ exp −κRð Þ; ð18Þ

where ω and κ are the density dependent real and imaginary parts of

the pole of L[βw(R)] with the smallest imaginary part, and γ is theresidue at the same pole.

The results of Eqs. (17) and (18) are valuable. In particular, Asakuraand Oosawa [1], who examined the depletion force at low densities ofthe suspending fluid have found that βw(0) = −3Dη/2d, where η =(π/6)ρd3 is the volume fraction of suspending fluid particles of diameterd1 = d2 ≡ d. We can see that in the low density limit, Eq. (17)reduces to − (3/2)δ2D that is exactly the result of Asakura andOosawa. At higher densities, however, it is always lower than thevalue − 1.4545δ2D predicted by the DFT approach [10], i.e., it overesti-mates the strength of depletion attraction between two macrospheresroughly by a factor (1 − δ3)−2. This is the penalty for using the PYapproximation to treat the fluid–fluid and fluid–macrosphere correla-tions. However, beyond the contact x = 0, as the separation betweenmacrospheres increases, the interaction energy calculated according toEq. (15), approaches the DFT data. Particularly, as we can see fromFig. 2, beyond the depletion region, defined by the position of themain repulsivemaxima in the energy profile, both results are practicallyindistinguishable. This is an expected result, since, as we already noted[37,40], both equations have common roots.

3.2. Two flat surfaces in a hard-sphere solvent

The asymptotic term (Eq. (18)) for the interaction energy w(R)together with Derjaguin construction (Eq. (13)) and (Eq. (14)), havebeen used recently [41] to derive an analytical expression for the filmenergy per unit area W(H) and disjoining pressure Π(H) exerted ontwo flat surfaces by a hard-sphere fluid film. In the case of the disjoiningpressure this expression has the form,

Π Hð Þ ¼ −p; 0 ≤ H b dΠ1 cos ωH þ ϕð Þ exp −κHð Þ; H ≥ d;

�ð19Þ

where d is the diameter of the suspending fluid particles while all otherparameters (Π1, ϕ, ω, κ) are fitted as cubic polynomials in terms of theparticle's volume fraction [41] and p refers to the bulk osmotic pressureof the suspending fluid.

Fig. 3. Disjoining pressure Π(H) between two plane parallel surfaces suspended in amonodisperse fluid composed of hard-sphere particles of diameter d and volume fractionη=0.314. The solid lines correspond to the results of parametrized expression (Eq. (19)).The symbols are the Monte Carlo simulation data due to Wertheim et al. [42].

Fig. 4. Interaction energyw(x) between twomacrospheres of diameter 3.5 μm suspendedin a monodisperse fluid of hard-sphere particles. The results correspond to threesuspending fluids that differ by particle diameters, namely: 0.25 nm, 2.5 nm and 20 nm,respectively. The particle volume fraction was fixed at 30% in each case. All curves havebeen computed using parametrized equation [10] that is based on the asymptotic form(Eq. (18)). The thick solid line shows the energy of van der Waals attraction which isreduced to the adhesion forces, acting between two macrospheres, with a Hamakerconstant assumed to be 10−20 J that is a typical value for latex particles interacting acrosswater [29].

36 A. Trokhymchuk, D. Henderson / Current Opinion in Colloid & Interface Science 20 (2015) 32–38

Fig. 3 shows the disjoining pressureΠ(H) exerted by a hard-spherefluid film that is calculated using expression (19) and is comparedagainst Monte Carlo simulation data obtained for the same systemsetup by Wertheim et al. [42].

The disjoining pressure expression (Eq. (19)) is parametrized tosatisfy with some known exact relations for a confined hard-core fluidand is designed to be easily implemented in the calculations of colloidalfilm properties in the various practical applications, e.g., to examine therole of the structural disjoining pressure in the movement of a three-phase contact line [43] that is directly related to the spreading ofnanofluid on solid surface [44] and to the removal of a pollutant froma solid surface by the action of a surfactant solution [45].

4. Outlooks

Despite the success in the understanding of the molecular contribu-tion to the excluded volume interaction, the statistical description of thedepletion interaction mediated by supramolecular solutions (suspen-sions of colloids, micelles, biological macromolecules in molecularsolvent, etc.) still usually ignores the contribution of the molecularsolvent as the discrete nature of solvent is disregarded by using theso-called “primitive” models where molecular solvent is treated as acontinuum suspending background. An appropriate model of thesuspending medium in such a case would be a binary mixture of hard-sphere particles with high diameter asymmetry. The presence in thesuspending fluid of the species with two very distinct dimensionscould affect significantly both the depletion as well as the structuralscenarios observed for a single-component suspending fluid [46].

4.1. Primitive estimate of the effect of solvent bidispersity

Therefore, let us analyze first on the same scale the interactionenergy induced by the individual suspending media, each composedof monosized hard-sphere species but with significantly differentdiameters. Fig. 4 shows a set of three interaction energy profiles thatcorrespond to three different suspending media characterized bydiameters: d = 0.25 nm, 2.5 nm and 20 nm, respectively. The volumefraction of suspending fluid particles in each case was maintainedconstant at the value of 30%. The macrosphere diameter D is chosen tobeD=3.5 μm that mimics the silica sphere often used in themeasure-ments [47]. Obviously, all three energy profiles are qualitatively quite

similar, i.e., all curves show a monotonic depletion attraction for therange of separations smaller than the corresponding fluid particlediameter d. Each energy profile exhibits a repulsive barrier at separa-tions of about one corresponding depletant diameter d and a dampingoscillatory structural repulsion if the separation exceeds that diameter.At the same time, quantitatively all three interaction energy profilesare quite different in magnitude as well as in range.

Since the volume fraction of suspending fluid in Fig. 4 is fixed, thedifferences that are observed should be attributed to the differences inbasic length scale, i.e. in the suspending fluid particle diameter foreach case. Indeed, the interaction energy that is mediated by asuspending fluid medium composed of the smallest particles (d =0.25 nm) is always higher in magnitude compared to that due tosuspending fluid media composed of larger particles (d = 2.5 nm or20 nm) by a factor proportional to the ratio of large-to-small fluid par-ticle diameters. However, each larger interaction energy is compressedto a narrower range of separations between surfaces such that propor-tionality factor is reversed to that for magnitudes. Generalizing, wecan state that suspending fluid-mediated interaction extends over sep-arations ranging from 3 to 5 suspending fluid particle diameters in eachcase (at least, for the volume fractions of the order of 30% that is typicalfor common solvents). Therefore, one could speculate, that even simplesuperposition of any of the pair of three curves in Fig. 4 to mimic thebidisperse suspending medium, will result in the energy profile that isqualitatively different from when the suspending media is treated as amonodisperse fluid.

Although the real systems differ from the hard-spheremodeling anda quantitative comparison with experimental measurement is some-what questionable, theory and observation agree remarkably well onthe main features of the suspending fluid-mediated oscillatory interac-tion betweenmacrosurfaces, including the periodicity of the oscillationsand their exponential decay. To make some conclusions regardingimportance of the suspending fluid-mediated contribution to the totalinteraction, the direct interaction between macrosurfaces must betaken into account. For comparison, Fig. 4 also shows the typical energyof the van der Waals forces that always are present in real systems andmust be superimposed with that due to suspending fluid if one is

37A. Trokhymchuk, D. Henderson / Current Opinion in Colloid & Interface Science 20 (2015) 32–38

interested in the total energy picture. Such a van der Waals interactionis always attractive with a magnitude proportional to the Hamakerconstant.

4.2. Solute spheres in a bidisperse hard-sphere solvent

The “primitive” analysis presented in Fig. 4 suggests the revision ofthe resulting suspending fluid-mediated interaction when the particleswith extremely different diameters are mixed and become the speciesof the same suspending media. So far, the smallest component (usuallythemolecular solvent) was not considered to consist of discrete specieswhen the effective interaction across supra-molecular fluid is analyzed.

Fig. 5 presents the results for the suspendingfluid-mediated interac-tion energy induced by a bidisperse suspending medium with asize ratio, small-to-large 1:10 as it is predicted by Eq. (15). Because ofthe large size asymmetry among the species of suspending fluid, thedepletion interaction scenario observed for two giant spheres in aone-component suspending fluid composed of only larger species, issignificantly affected when the fine species of suspending fluid aretaken into account. The gap between two giant spheres that is depletedof the large particles in a monodisperse fluid, becomes filled by the fineparticles in the case of a bidisperse fluid. Now the small species of diam-eter d1 exhibit stratification since they are confined by giant spheres ofdiameter D and the large suspending fluid species of diameter d2. Thischanges qualitatively the shape of both the interaction energy andforce profiles that now show fine oscillatory structure at separationsnear the contact of the giant spheres; these oscillations are governedby the diameter of small species.

5. Concluding remarks

The suspending fluid-mediated interactions have been extensivelystudied and so far are well understood in the case of a monodispersesuspending medium. Two basic features of interaction energy mediat-ed by a monodisperse suspending fluid, include: (i) the depletionattraction — as a result of the absence of suspending fluid species inthe space between macrosurfaces, and (ii) the structural repulsivebarrier — as a result of the layering of suspending fluid species in thegap between the macrosurfaces. The oscillatory behavior of interactionenergy at larger separations results from the layer-by-layer structuring

Fig. 5. Interaction energy w(x) between two macrospheres of diameter D that aresuspended in a bidisperse hard-sphere fluid composed of the fine species of diameterd1 = 0.01D and volume fraction η1 = 0.15 and the larger species of diameter d2 = 10d1and volume fraction η2 = 0.20.

of the suspending fluid particles in the gap between two macrosurfaceswhen the gap width exceeds one particle diameter.

The AO approach for the depletion interaction energy has been verypopular because it consists of a useful analytic expression. However,Eq. (7) that in lowdensity limit reproduces AO result, also yields a usefulanalytic expression although of the higher complexity. A Fortran codefor this expression is given in the Supplementary material (SM4) to arecent publication of Smith et al. [48]. With this code accompanied byEq. (10), a convenient generalization of the AO approach that is validat higher densities, is available. Additionally, we recall a simple analyti-cal expression of Ref. [41] resulting from Eq. (7) based on the largedistance asymptotic form (Eq. (18)).

The bidispersity of a suspending fluid, that is the case of practicalimportance, continues to be considered in a “primitive” manner whenthe smaller species of a suspendingmediumare treated as a continuum.In particular, the depletion attraction mechanism introduced byAsakura and Oosawa neglects the presence of the molecular solvent(or fine particles) in the space between macrosurfaces. However, fromwhat we discussed here it seems that the presence in the suspendingfluid of the species with two distinct dimensions affects significantlyboth the depletion as well as the structural scenarios observed for amonodisperse suspending fluid.

It is interesting to note that due to the large asymmetry in the size ofthe species that compose the suspending fluid, the impact of eachcomponent is split on the length scale. Due to this, it seems that tosome extent the effect of bidispersity can be treated as the superpositionof the results for two individual monodisperse suspending mediacomprised of the small and large particles, respectively. This lattersuggestion requires additional investigations.

References

[1] Asakura S, Oosawa F. On interaction between two bodies immersed in a solution ofmacromolecules. J Chem Phys 1954;22(7):1255–6.

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