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Confinement Effects on Phase Behavior of Soft Matter

Systems

Kurt Binder∗ Jurgen Horbach† Richard Vink‡ Andres De Virgiliis§

November 2, 2018

When systems that can undergo phase sepa-ration between two coexisting phases in thebulk are confined in thin film geometry betweenparallel walls, the phase behavior can be pro-foundly modified. These phenomena shall bedescribed and exemplified by computer simula-tions of the Asakura-Oosawa model for colloid-polymer mixtures, but applications to other softmatter systems (e.g. confined polymer blends)will also be mentioned. Typically a wall willprefer one of the phases, and hence the composi-tion of the system in the direction perpendicularto the walls will not be homogeneous. If bothwalls are of the same kind, this effect leads to adistortion of the phase diagram of the system inthin film geometry, in comparison with the bulk,analogous to the phenomenon of “capillary con-densation” of simple fluids in thin capillaries. Inthe case of “competing walls”, where both wallsprefer different phases of the two phases coexist-ing in the bulk, a state with an interface parallelto the walls gets stabilized. The transition fromthe disordered phase to this “soft mode phase”is rounded by the finite thickness of the film andnot a sharp phase transition. However, a sharptransition can occur where this interface gets

∗Institut fur Physik, Johannes Gutenberg UniversitatMainz, Staudinger Weg 7, 55099 Mainz, Germany. E-mail: kurt.binder@uni-mainz.de

†Institut fur Materialphysik im Weltraum, DeutschesZentrum fur Luft- und Raumfahrt (DLR), 51170 Koln,Germany. E-mail: juergen.horbach@dlr.de

‡Institut fur Theoretische Physik, Georg-August Universitat, Friedrich-Hund-Platz 1, 37077Gottingen, Germany. E-mail:vink@theorie.physik.uni-goettingen.de

§Instituto de Investigaciones Fisicoquimicas, UNLP,CONICET, Sucursal 4, Casilla de Correo 16, 1900 LaPlata, Argentina. E-mail: adevir@inifta.unlp.edu.ar

localized at (one of) the walls. The relation ofthis interface localization transition to wettingphenomena is discussed. Finally, an outlook torelated phenomena is given, such as the effectsof confinement in cylindrical pores on the phasebehavior, and more complicated ordering phe-nomena (lamellar mesophases of block copoly-mers or nematic phases of liquid crystals underconfinement).

1 Introduction

The current interest in the construction ofnanoscopic devices [1, 2, 3, 4, 5] demands abetter understanding of the phase behavior offluids confined in pores or slits of nanoscopiclinear dimensions [6, 7, 8, 9, 10, 11, 12]. Knowl-edge on the phase behavior of confined fluidsis a prerequisite to understand their dynamics[13, 14, 15], as well as for the analysis of flowthrough very thin capillaries [16, 17], nanoscalecapillary imbibition [18, 19], and related mi-crofluidic or nanofluidic devices.

Obviously, an interplay must be expected be-tween surface effects on the fluid due to the con-fining walls, such as adsorption [20, 21, 22, 23],formation of wetting (or drying) layers [24, 25,26, 27], and finite size effects [28, 29, 30] due tothe finite width of the capillary. However, un-derstanding the nanoscopic confinement of realfluids consisting of small molecules is very diffi-cult due to additional effects, resulting from thelateral variation of the wall potential caused bywall roughness or even the atomistic corrugation[31, 32] of the wall.

While there has been an enormous activity to

1

study theoretically and by computer simula-tion confinement effects on simple fluid mod-els such as the Ising lattice gas model [11] orsimple Lennard-Jones systems [10, 12] and richpredictions from phenomenological theories areavailable as well [7, 8, 9, 10, 11, 12, 33, 34,35, 36, 37, 38, 39], it is difficult to find perti-nent experiments to which such work could becompared. However, it is much more promis-ing to study confinement effects on soft mattersystems: due to the mesoscopic length scalesof the particles that one encounters when onestudies mixtures of polymers and colloids [40]or polymer blends [41, 42], effects due to theatomistic corrugation of the walls are much lessimportant; also the large size of colloidal parti-cles enables more detailed experimental obser-vations; e.g. individual particles can be trackedthough real space in real time using confocal mi-croscopy [43] and interface fluctuations in mix-tures of colloids and polymers can be directlyobserved [44, 45]. Moreover, colloids are modelsystems for the study of phase behavior, sinceby changing suitable parameters the strengthand range of effective interactions can be variedover a wide range [46, 47, 48]. Also the inter-action of colloidal particles with the confiningwalls can be tuned, e.g. by coating the wall witha polymer brush [49, 50, 51, 52] and controllingthe polymer-wall interaction via variation of thegrafting density and/or chain length of the an-choring flexible polymer [52, 53]. In particular,for colloid-polymer mixtures both the radius ofthe (spherical) colloidal particles and the size ra-tio between colloids and polymer coils controlsthe location of the critical point where the phaseseparation in a colloid-rich and a polymer-richphase sets in [40].

Similarly, also blends of long flexible polymersare a very suitable model system to study theeffect of confinement in a thin film geome-try on phase separation experimentally as well[41, 42, 54]. Again, already the location of thecritical temperature of phase separation in thebulk can be varied over a wide range, by suit-able choices of the polymeric species, and theirchain lengths [55, 56, 57]. In addition, charac-teristic lengths of the problem such as the cor-relation length of composition fluctuations [57],

the (intrinsic) interfacial width between coex-isting phases [57], etc., are much larger thaninteratomic distances, and hence also for thesesystems experimental probes are available whichwould lack sensitivity for small molecule sys-tems. E.g., for the study of the anomalousbroadening of interfaces depending on the filmthickness [41, 42], which is one of the char-acteristic signatures of the “soft mode phase”[36, 37] in a system with “competing walls” [38],a nuclear-reaction based depth profiling method[54] was used. This method can resolve thevery wide interfaces in such soft matter systems,while it would be unsuitable for the much nar-rower interfaces in mixtures of small molecules.

As is evident from this introductory discus-sion, and the extensive literature that has al-ready been quoted, the subject is extremelyrich, and comprehensive coverage could fill awhole book. Therefore the scope of the presentreview necessarily must be more narrow. Weshall focus in this review almost exclusivelyon confinement effects of colloid-polymer mix-tures [58, 59, 60, 61, 62, 63]. Only spher-ical colloidal particles shall be discussed, al-though related phenomena can be studied alsofor mixtures of polymers with rod-like colloids[64]. Although extensive work has been done formodels of polymer blends, both using the self-consistent field theory [65, 66, 67] and simula-tions [65, 68, 69, 70], we shall not consider thiswork here, but draw attention to recent reviews[71, 72]. Also, we shall not attempt to reviewthe theory of wetting phenomena [24, 25, 26, 27]and scaling theories of capillary condensation[11, 33] and interface localization transitions[36, 37, 38, 70] but rather refer the reader toanother thorough review [11].

In Sec. 2, we shall briefly recall work on phasebehavior of the model of Asakura and Oosawa(AO) [73] and Vrij [74], where colloids simplyare described as hard spheres which may nei-ther overlap with each other nor overlap withpolymers, while the latter may overlap witheach other with no energy cost, in the bulk[75, 76, 77, 78, 79, 80, 81, 82, 83]. In Sec. 3we shall discuss the phase behavior of this AOmodel when it is confined [58, 59, 60, 61, 62]between symmetrical walls a distance D apart,

2

paying attention to the shift of the critical pointas a function of film thickness D, and to thechange of the critical behavior. Sec. 4 then de-scribes the behavior encountered for asymmet-ric walls [63], where it is also shown that byvariation of the conditions at the walls one cangradually crossover from this interface localiza-tion transition to a transition which is of capil-lary condensation type. Sec. 5 then presentsa summary of the results reviewed here, andgives an outlook on related findings in othersystems, as well as to more complicated phe-nomena where the order parameter characteriz-ing the transition is not a simple scalar quantity(as it is for gas-liquid or liquid-liquid type phaseseparation).

2 Liquid-liquid demixing

for the Asakura-Oosawa

(AO) model in the bulk

In the AO model [73, 74] colloids are describedas hard spheres of radius Rc, and hence thepotential between two colloidal particles at dis-tance r from each other is

Ucc(r) = ∞ (r < 2Rc), Ucc(r) = 0 (else) .(1)

Similarly, polymers are described as soft spheresof radius Rp. Remembering that long polymerchains with N subunits have a radius Rp ∝ Nν

with ν ≈ 0.59 in good solvent conditions [56] orRp ∝ N1/2 in Theta solvents [56], the densityρp = N/R3

p of monomers of a chain inside itsown volume is very small, and hence polymercoils can interpenetrate each other with a freeenergy cost of a few kBT (with kB the Boltz-mann constant and T the temperature) [84]. Inthe AO model, this free energy cost is neglected,and the polymers are treated like particles in anideal gas, Upp(r) = 0 irrespective of distance.But, of course, polymers cannot penetrate intothe colloidal particles, and hence

Upc(r) = ∞ (r < Rc +Rp), Upc(r) = 0 (else).(2)

As is well-known [40], the polymers cause an(entropic) depletion attraction between the col-

loidal particles, and as a result, an entropy-driven phase separation occurs, if the volumefractions ηc, ηp of colloids and polymers are suf-ficiently high (Fig. 1). Here ηc, ηp are definedin terms of the volume V of the system and thenumbers of colloids and polymer, Nc and Np,respectively, by

ηc =4π

3R3

cNc/V , np =4π

3R3

pNp/V , (3)

and Rc = 1 will henceforth be chosen as unitof length. Since both ηc, ηp are densities ofextensive thermodynamic variables, it is usefulto carry out a Legendre transform to an inten-sive thermodynamic variable, where the chemi-cal potential µp of the polymers [or their fugac-ity zp = exp(µp/kBT )] is used. It is customaryto use instead of µp or zp the so-called “polymerreservoir packing fraction” ηrp,

ηrp = zp(4π

3

)

R3p (4)

Eq. (4) would be just the volume fraction ofpolymers in the absence of any colloids, sincesuch a system simply is an ideal gas of polymers.It is clear that the model defined by Eqs. (1), (2)is a drastic simplification of reality, but in qual-itative respects it is remarkably accurate [40].While various more realistic extensions of theAO model have been considered [79, 84, 85, 86,87, 88, 89, 90, 91, 92], and sometimes betteragreement with experiments [44, 93] is obtained,we disregard such extensions here because inpractice there are many additional effects (suchas charges on the colloidal particles [94, 95], ad-sorption of polymers on the colloids [96], etc.)that make a quantitative comparison with ex-periment elusive.In early simulation work on the AO model[77, 78] a wider range of volume fractions ηc, np

(and a much wider range of ηrp) was studied, butonly a much more limited accuracy than shownin Fig. 1 was obtained. On the basis of this work[77, 78], it was concluded that the agreementbetween simulations and the mean-field theoryof Lekkerkerker et al. [75] is excellent. Fig. 1demonstrates, however, that the relative devia-tion between the actual value for ηrp at the criti-cal point, ηrp,cr = 0.766±0.002, deviates from its

3

mean-field prediction [75] by about 30%. Thisdeviation, in fact, is relatively larger than corre-sponding deviations between mean field theoryand accurate simulation results for lattice gasmodels [97], Lennard-Jones fluids [98], etc. Inretrospect, this large deviation between mean-field theory [75] and accurate simulation resultsfor colloid-polymer mixtures [80, 81, 82] is notsurprising, since on the length scale of a colloidalparticle the depletion attraction has a very shortrange, and the large absolute size of colloidalparticles in this context is not relevant: it wouldbe wrong to infer that colloids should behavemean-field like.

Being interested in the changes in phase behav-ior due to confinement between walls that area distance D ≫ Rc(= 1) apart, relatively smallchanges must be expected, of course. For ananalysis of these changes, and in particular fora study how bulk behavior in the limit D → ∞is approached, a very good accuracy of the sim-ulation data is absolutely crucial. Thus, it isworthwhile to briefly recall how results such asthose shown in Fig. 1 can be obtained, since themethods for the study of the confined systems[61, 62, 63] are closely related to those used inthe bulk [80, 81, 82].

We start this recollection by emphasizing thatfor studying liquid-vapor type phase equilibriathe grand-canonical (µV T ) ensemble of statis-tical mechanics is the best choice [98], sinceit avoids problems due to slow relaxation ofliquid-vapor interfaces that hamper the use ofthe canonical ensemble [99, 100]. Also nearthe critical point the problem of critical slow-ing down [101] is somewhat less severe in thegrand-canonical ensemble [98, 99, 100], andthe inevitable finite size effects are relativelyeasy to handle by finite size scaling methods[28, 29, 30, 102, 103], unlike the popular Gibbsensemble [104, 105]. So the task of the simula-tion is to vary the chemical potential µ of thecolloids at fixed ηrp (as the phase diagram, Fig. 1,suggests, ηrp is analogous to inverse temperaturefor ordinary vapor-liquid type transitions [98],where vapor-liquid phase separation is driven byenthalpic rather than entropic forces. Of course,in thermal equilibrium the average colloid frac-tion 〈ηc〉, which is the variable thermodynam-

ically conjugate to µ (apart from a normaliza-tion factor, see Eq. 3), increases monotonouslywith µ even when the two-phase coexistenceregion is crossed, and in the 〈ηc〉 vs. µ curvehence no singularity shows up for any finite lin-ear dimension L: only in the thermodynamiclimit (where L → ∞) this “isotherm” devel-ops at µ = µcoex a perpendicular part, where〈ηc〉 jumps discontinuously from ηVc (vapor) toηLc (liquid). However, nevertheless phase coex-istence is easily recognizable also in a finite vol-ume simulation, when the colloid volume frac-tion distribution P (ηc) is sampled [98, 99, 100].In the regime ηVc ≤ 〈η〉 ≤ ηLc , P (ηc) has a dou-ble peak structure, and for µ = µcoex both peakshave equal weight (“equal area rule” [106, 107]).

In order to carry out this program, two obstaclesneed to be overcome: (i) in order to sample therelative weights of the two peaks of P (ηc), thepeak near ηVc representing the vapor-like phaseof the colloid-polymer mixture and the peaknear ηLc , the liquid-like phase, the system needsto cross many times a region of very low prob-ability near ηd = (ηVc + ηLc )/2. This problem,however, can be very efficiently solved by suc-cessive umbrella sampling [108]. Fig. 2 shows, asa typical example, distributions P (ηc) that spanalmost 30 decades. (ii) The second obstacle isthe fact that the polymer volume fraction, in thepolymer-rich phase, can be very high (exceedingunity, since the polymers are allowed to overlapwith no energy cost). Insertion of a colloid par-ticle at a randomly chosen position, which is oneof the Monte Carlo (MC) moves that one needsto carry out in grand-canonical Monte Carlosimulations, almost always will be rejected: so anaive implementation of a grand-canonical MCsimulation for unfavorable parameters is boundto fail utterly. However, this problem also couldbe overcome, by the invention of a compositeMC move, where in a spherical region with someproperly chosen radius rc a randomly selectedchosen number nr of polymers is taken out andonly then insertion of a colloid is attempted (thereverse move also exists and is constructed suchthat the detailed balance principle [99, 100] isfulfilled) [81, 82].

We now return to the observation of Fig. 2, thata high free energy barrier ∆F (choosing units

4

where kBT = 1) exists, which is independent ofηc in a broad regime of ηc around the compo-sition of the rectilinear diameter ηd. The inter-pretation of this fact is that the system in thisregion is in a state with two domains, separatedby two domain walls, oriented perpendicular tothe z-direction, and connected into itself by theperiodic boundary conditions. This is also con-firmed by direct inspection of the configurationsof the system. Hence [109]

∆F = 2L2γLV, L → ∞ , (5)

where γLV is the interfacial tension betweenliquid- and gas-like phases, and L2 the inter-facial area. Thus, estimating ∆F for a se-ries of cross-sectional areas L2 of the simula-tion box and extrapolating the result for γLVto the thermodynamic limit has become a stan-dard method for the MC estimation of interfa-cial free energies [99, 100]. Fig. 3 shows typ-ical results for the reduced interfacial tensionplotted vs. the order parameter ηLc − ηVc andcompares them to density functional theory pre-dictions [110]. These simulation results for γLVare also consistent with a capillary wave anal-ysis [83]. Note that the coexistence densitiesηVc , η

Lc do approach the predictions from mean

field theory rather fast (Fig. 1), for ηrp ≥ 1.0 thedifferences are practically invisible, however, nosuch convergence is seen for the interfacial ten-sion (Fig. 3). The reason for the strong discrep-ancies in Fig. 3 is not clear.We now comment on the treatment of finite sizeeffects. If one naively would take the values ofηc where P (ηc) has its two peaks as estimates forηVc and ηLc also in the critical region, one obtainsresults as shown in Fig. 4: For ηrp ≥ 0.79 theseestimates are independent of the linear dimen-sion of the simulation box, but for ηrp < 0.79 sys-tematic finite size effects appear. E.g., for ηrp =0.76 the difference ηLc − ηVc decreases systemati-cally with increasing L. While for ηrp > ηrp,cr thisdifference for L → ∞ converges to a nonzeroresult, for ηrp ≤ ηrp,cr it ultimately vanishes.While a naive inspection of Fig. 4 does not allowto estimate ηrp,cr, such an estimate can be ob-tained reliably from finite size scaling methods[28, 29, 30, 87, 88, 89, 90, 100, 102, 103]. Choos-ing µ = µcoex(η

rp) from the equal area rule, as is

done in Figs. 1, 4, we define an order parameterm as m = ηc − 〈ηc〉 and define moments 〈|m|k〉(k being integer) from the distribution P (ηc),

〈|m|k〉 =

1∫

0

dηc |m|kP (ηc) . (6)

Defining then the fourth order cumulant U4 as[97, 103].

U4 = 〈m2〉2/〈m4〉 (7)

we can invoke the result that U4 tends towardsunity for ηrp > ηrr,cr as L → ∞, while U4 tendsto 1/3 for ηrp < ηrp,cr, since ultimately the distri-bution P (ηc) in the one-phase region must be-come a single Gaussian centered at 〈ηc〉 [103].For ηrp = ηp,cr,however, U4 tends to a nontrivialbut universal value U∗

4 (U∗

4 ≈ 0.629 in d = 3 di-mensions while U∗

4 ≈ 0.856 in d = 2 dimensions[111]). Consequently, plotting U4 versus ηrp fordifferent L one expects a family of curves thatintersect at ηrp = ηp,cr in a common intersectionpoint, if L is large enough so that corrections tofinite size scaling are negligible, and using thismethod (or an analogous reasoning [112] for themoment ratio M = 〈m2〉/〈|m|〉2 [80, 81] whichshould yield a universal intersection in d = 3 atM∗ = 1.239 [113]) one finds the estimate of ηrp,crincluded in Figs. 1, 4.A further consequence of finite size scaling [28,29, 30, 97, 98, 99, 100, 102, 103] is the fact thatthe moments 〈|m|k〉 are homogeneous functionsof the two variables L and t = ηrp/η

rp,cr − 1,

〈|m|k〉 = L−kβ/νMk(x), x = tL1/ν , (8)

where β and ν are the critical exponents of theorder parameter Mc and correlation length ξ,respectively,

Mc = Btβ , ξ = ξt−ν . (9)

In Eq. (8), Mk(x) is a scaling function, and Band ξ are critical amplitudes. For the univer-sality class of the d = 3 Ising model [114], theexponents are [97, 115, 116]

β ≈ 0.326, ν ≈ 0.630, (10)

which differ from the corresponding mean-fieldresults [114]

βMF = 1/2, νMF = 1/2 . (11)

5

Taking the estimates for the exponents[Eq. (10)] and using ηrp,cr = 0.765 we can re-plot the data of Fig. 4 in scaled form (Fig. 5),and indeed the data collapse rather well on amaster curve, as implied by Eq. (8). If we useEq. (11) instead, no such data collapsing is ob-tained. This result shows that finite size scal-ing holds, and the AO model also falls in thed = 3 Ising universality class, as one might haveexpected. Moreover, the straight line behaviorseen on the log-log plot for large x not only im-plies that the data indeed are compatible withthe power law, Mc = Btβ , but also the criticalamplitude B can be estimated with reasonableaccuracy, B = 0.27± 0.02 [83]. This power lawactually has been included in Fig. 4 for ηrp nearηrp,cr. It results from Eq. (8) as the asymptoticbehavior for L → ∞.

As is evident from the insert of Fig. 1, the fluc-tuations that are ignored by mean-field theory[75] have two effects: one effect is that the criti-cal point ηrp,cr is shifted upward (the compatibil-ity of the colloid-polymer mixture is enhanced),and the coexistence curve is flattened near thecritical point [according to mean-field theory,Eq. (11), it is a simple quadratic parabola].

A similar discussion can be given for the inter-facial tension, γLV (Fig. 3), which is found tovary as [83]

γLV = γ tµ, µ = 1.26, γ ≈ 0.26±0.02, (12)

while mean-field theory would imply µ = 3/2[114]. Vink et al. [83] have also analyzed thecritical behavior of susceptibilities at both sidesof the transition and studied the rectilinear di-ameter ηd, as well as a few critical amplituderatios. All these analyses did confirm the Isingcharacter of the transition, indicating that theIsing critical region in fact is remarkably wide.Mean-field theory [75] is only reliable very faraway from criticality.

3 Confinement by Sym-

metric Walls: Evi-

dence for Capillary-

Condensation-Like Be-

havior

In this section we consider colloid-polymer mix-tures in a L × L ×D geometry, where confine-ment is effected by two identical walls a distanceD apart. In the simulations, we apply periodicboundary condition in the x and y-directionsparallel to the walls, and again the strategy willbe to carry out an extrapolation to the thermo-dynamic limit via a finite size scaling analysis.

If one simply uses hard walls for both colloidsand polymers, as done in [61], one encoun-ters a very pronounced depletion attraction be-tween the colloids and the walls, giving rise to avery strong “capillary condensation”-like shift[10, 11] of the coexistence chemical potentialµcoex(η

rp) of the colloids. It is hence convenient

to apply in addition a square-well repulsive po-tential

Ucw(h) = ε, Rc < h < 2Rc, Ucw(h > 2Rc) = 0,(13)

with h the distance of a colloidal particle fromthe (closest) wall. Of course, Ucw(h ≤ Rc) = ∞and Upw(h ≤ Rp) = ∞, since neither colloidsnor polymers are allowed to penetrate into thewall.

If one considers very large ε, colloids are ex-cluded from the close vicinity of the walls, andan effective attraction of the polymers to thewalls would result. As a consequence, “capillaryevaporation” is expected rather than “capillarycondensation” (i.e., close to phase coexistencein the bulk the capillary prefers the vapor-likephase rather than the liquid-like phase of thecolloid-polymer mixture). Schmidt et al. [59]presented a (somewhat qualitative) evidence forthis phenomenon.

Since the finite size thickness D limits growthof the correlation length ξ of volume fractionfluctuations near the critical point in the z-direction perpendicular to the confining wall, adivergence of ξ as described in Eq. (9) is only

6

possible along the x- and y-directions parallel tothe walls. Therefore, the phase transition whichcan take place is a phase separation in lateral di-rections (x, y) only, between colloid-rich and col-loid pure phases. As a consequence, ultimatelythis transition should belong to the universalityclass of the two-dimensional Ising model [114],and the critical exponents are

β = 1/8, ν = 1, µ = 1, (14)

instead of those quoted in Eqs. (10), (12). How-ever, this two-dimensional critical behavior pre-vails only when ξ has grown to a size much largerthan D: if ξ ≪ D the behavior is still close tothree-dimensional, and when ξ and D are of thesame order a gradual crossover between the twotypes of critical behavior occurs.These crossover phenomena make the analysisof the simulations somewhat more difficult. Forany finite value of L the “raw data” estimatesfor ηVc , η

Lc are qualitatively similar, irrespective

of D (Fig. 6). Again pronounced “finite sizetails” occur for these estimates in the vicinityof ηrp,cr, i.e., for any finite L one finds that ηVcand ηLc as estimated from the peak positions ofP (ηc) fail to merge at ηrp,cr, but rather continuefurther into the one-phase region, as in the bulk(Fig. 4). When one then plots U4 vs. ηrp fordifferent choices of L, searching for a universalintersection point, one rather finds that the in-tersection points are somewhat scattered over aregion of values for ηrp (Fig. 7). In addition, thisintersection does occur neither at the theoreti-cal value for U∗ for the d = 2 universality classnor at the U∗ for d = 3, but rather somewherein between. These findings are a consequenceof the gradual crossover in critical behavior al-luded to above. While for D = 3 both ν andU∗ are rather close to the theoretical d = 2values, for D = 10 both ν and U∗ are abouthalf way between the d = 2 and d = 3 values.However, these numerical results do not haveany fundamental significance; they only meanthat the larger D the closer ηrp,cr needs to be ap-proached, to be in the region where ultimatelyξ ≫ D and hence the correct asymptotic criti-cal behavior (which is always two-dimensional,for any finite value of D) can be seen.For the case D = 5, ε = 0 a very careful

analysis has been performed [61], applying anovel variant of finite scaling which does notimply any bias on the type of critical exponents[117, 118, 119]. Fig. 8 shows that the resultingorder parameter can be fitted over some rangeindeed by an effective exponent βeff = 0.17,which is in between the d = 2 and d = 3 values(0.125 < βeff < 0.326), but a correct interpre-tation of this finding is that a log-log plot ofthe order parameter vs. t exhibits a slight cur-vature, spread out over several decades. Onlyfor t → 0 can the d = 2 value (β = 0.125) beexpected to be seen; for larger t the slope βeff onthe log-log plot increases systematically (but itdoes not reach the d = 3 value, since for t ≥ 0.1noncritical saturation effects come into play).While the critical behavior of thin films is con-trolled by the d = 2 critical exponents, a differ-ent answer results when one considers the shiftof the critical point relative to the bulk [33, 34]:this shift is controlled by three-dimensional ex-ponents only, namely

∆ηrp,cr(D) ≡ ηrp,cr(D)− ηrp,cr(∞)

∝ D−1/ν , D → ∞ (15)

∆µcoexcr (D) ≡ µcoex

cr (D)− µcoexcr (∞)

∝ D(∆1−∆)/ν , D → ∞. (16)

Here ∆ ≈ 1.56 [97, 115, 116] is the so-called“gap exponent” which characterizes the bulkequation of state near criticality, and ∆1 ≈ 0.47[120, 121, 122, 123] its surface analog. Figure9 shows that the AO model is compatible withthese predictions, even though only rather smallfilm thicknesses were accessible to the simula-tion (the largest film thickness included in Fig. 9is only for D = 10 colloid diameters).Note that for ηrp > ηrp,cr the asymptotic behaviorof the shift of the colloid chemical potential atphase coexistence is not given by Eq. (16), butby the simpler “Kelvin equation” [10, 35]

∆µcoex(D) = µcoex(D)− µcoex(∞)

∝ 1/D, D → ∞, ηrp > ηrp,cr.(17)

Fig. 10 shows that the data of Vink et al. [62]are compatible with this equation as expected.We emphasize that in mean field theory onecould not discuss the crossover between two-

7

and three-dimensional critical behavior, sinceβMF = 1 irrespective of dimensionality [114],and also the mean-field predictions for the shiftof the critical point [Eqs. (15), (16)] would bedifferent from what was observed [62], since1/νMF = 2 instead of 1/ν ≈ 1.59, and (∆MF

1 −∆MF)/νMF = −2 instead of (∆1 − ∆)/ν ≈−1.73. However, mean-field theory does repro-duce the Kelvin equation, Eq. (17), and in anycase mean-field results for our confined filmswould be desirable. We note that some mean-field results as well as Monte Carlo results areavailable for q = 1 [58, 59]; however, the accu-racy of these Monte Carlo data was too limitedto allow for a comprehensive test of theoreticalpredictions, as reviewed above, and hence thesestudies [58, 59] are not discussed further here.

We conclude this section by discussing thestructure of the coexisting phases in the thinfilm in more detail. Already snapshot pictures[Fig. 11] show that in the z-direction the com-position can be inhomogeneous. For ε = 0colloidal particles are enriched at the walls inthe vapor-like phase, while for ε = 2 poly-mers are enriched at the walls in the liquid-like phase. However, it would be wrong to con-sider these enrichment layers as wetting layers[24, 25, 26, 27]: wetting layers are macroscopi-cally thick, and cannot occur in a thin film ge-ometry [11]. One should also recall that wet-ting at the surface of a semi-infinite system oc-curs at bulk coexistence, while coexistence inthe thin film deviates from bulk coexistence {cf.Eqs. (16), (17), and Figs. 9a, 10}. Fig. 12 showsdensity profiles across the thin film for severaltypical choices of parameters. One recognizesthat the colloid density in the liquid -like phasenear the walls shows a pronounced layering ef-fect, while the polymer density in the vapor-likephase lacks a corresponding effect. This findingis expected, since layering is a consequence ofthe repulsive interactions among the particles.While for D = 10 the films do reach homoge-neous bulk-like states in their center, for D = 3and D = 5 (not shown) the behavior stays in-homogeneous throughout the film.

4 Confinement by Asym-

metric Walls: Evidence

for an Interface Localiza-

tion Transition

By asymmetric walls one can realize a situa-tion that one wall attracts predominantly col-loids and the other wall attracts polymers. Asdiscussed in the previous section, hard walls at-tract colloids via a depletion mechanism; butcoating the wall by a polymer brush undersemidilute conditions, one may cancel this de-pletion attraction partially or completely, andalso reach a situation where polymers get at-tracted to the wall. This situation is qualita-tively modelled by a step potential of heightε, acting on the colloids only {Eq. (13)}. Anasymmetric situation occurs e.g. if the left wallis a hard wall but on the right wall the ad-ditional potential described by Eq. (13) acts,see Fig. (13). In drawing the schematic phasediagrams, we have assumed that for a semi-infinite system colloid-polymer mixtures exhibitcomplete wetting [11, 24, 25, 26, 27] over awide range near the critical point, namely forηrp,cr ≤ ηrp ≤ ηrp,w, while for ηrp > ηrp,w “in-complete wetting” (i.e., a nonzero contact an-gle of a droplet) would occur. This assump-tion is corroborated by density functional cal-culations [124] and Monte Carlo simulations[105, 125]. Since no “prewetting transition”[11, 24, 25, 26, 27] was found, the wetting transi-tion presumably is of second order, and this wasassumed drawing the phase diagrams of Fig. 13,since this greatly simplifies the theoretical anal-ysis. For the case of symmetrical mixtures oflong flexible macromolecules, the influence ofprewetting phenomena on the phase diagram ofthin confined films has been thoroughly investi-gated [11, 39, 65, 66, 67, 68, 69, 70, 71, 72], andit has been shown that typically a phase dia-gram with two critical points and a triple pointcan be expected.

For asymmetric walls an interface localizationtransition may occur, and this situation is ex-plained qualitatively in the right part of Fig. 13.If the strength of the attraction of the colloids

8

of the left wall is of the same order as thestrength of the attraction of the polymers to theright wall, ηr,rightp,w and ηr,leftp,w will be rather closeto each other and both exceed ηrp,cr distinctly.Then for ηrp > ηrp,cr the left wall will alwaysbe coated with colloids, the right wall will al-ways be coated with polymers. In other words,we expect an interface between the colloid-richphase on the left and the polymer-rich phaseon the right. When µ is small enough {i.e,µcoex(∞)−µ is large enough}most of the systemis in the polymer-rich phase (shown schemati-cally as BIIb in Fig. 13) but when µ increasesa transition takes place to a state where mostof the film is in the colloid-rich phase (stateBIIa). For ηrp > ηrp,cr(D) this transition isa sharp (first-order) phase transition, i.e. theinterface jumps from a state localized at theleft wall to a state localized at the right wall.For ηrp = ηrp,cr(D) this transition is of first or-der, while for ηrp < ηrp,cr(D) the transition is asmooth gradual transition (near the broken linein Fig. 13). Note, however, that this transitionbecomes sharper and sharper as D increases,but a true phase transition appears only in adiscontinuous manner in the limit D → ∞ [11]:then the broken line in Fig. 13 coincides with theline µ = µcoex(∞) ending at ηrp,cr , and ηrp,cr(D)does not converge to ηrp,cr but rather we haveηrp,cr(D → ∞) = ηr,leftp,w (for the situation drawnin Fig. 13).

For the states along the broken curve in Fig. 13the system is essentially inhomogeneous, thereexists a thick domain of colloid-rich phase inthe left part of the film, and a thick domainof polymer-rich phase in the right part, sepa-rated by a “delocalized” interface in the cen-ter of the film [11, 36, 37, 38, 39]. Snapshotpictures of the system indeed readily confirmsuch a scenario (Fig. 14), as well as the densityprofiles across the thin film (Fig. 15). Fig. 15ashows the profile for ηrp = 0.7 < ηrp,cr, i.e. astate in the one phase region of the bulk. Onerecognizes that the colloid concentration is en-hanced near the hard wall, as expected fromthe depletion attraction. Near the other wall atz = D = 10, the colloid concentration is some-what depressed, but the polymer concentrationis clearly enhanced. But in the center of the thin

film both profiles are roughly constant, as ex-pected for bulk-like behavior. In fact, for largeD we expect that the surface enhancement (orreduction, respectively) decays with z accordingto an exponential relation, exp(−h/ξ), where his the distance from the closest wall and ξ thebulk correlation length [120]. Figure 15b showsthe profiles at ηrp = 0.95, which exceeds the bulkcritical value ηrp,cr, but still is smaller than thecritical value ηrp,cr(D) of the confined system.Now, the profiles are very different from thoseof Fig. 15a: phase separation in a colloid-richand a polymer-rich phase has occurred, withthe interface position (estimated from the in-flection point of the polymer volume fractionprofile ηp(z), for instance) being located in thecenter of the film. This situation corresponds tothe snapshot in Fig. 14b. The interfacial profileresembles that of an interface between bulk co-existing phases, broadened by capillary waves[68, 83]. Finally, the cases ηrp = 1.2 (Fig. 15c,d)refer to the two-phase region of the film. The in-terface either is located near the hard wall, cor-responding to the polymer-rich phase of the film(Fig. 15c: this case corresponds to the snapshotshown in Fig. 14a), or near the wall that attractsthe polymers (Fig. 15d). Note that along thetransition line drawn schematically in Fig. 13(right part), there occurs lateral phase separa-tion between the states corresponding to thesetwo types of profiles, Fig. 15c and Fig. 15d,which hence can coexist with each other in athin film (and then are separated by an inter-face running from the right wall towards the leftwall).

Figure 16 shows the corresponding phase di-agrams for two film thicknesses, D = 5 andD = 10, varying the strength ε of the poten-tial [Eq. (13)] at the right wall. One sees thatwith increasing ε the critical points and thewhole coexistence curves are shifted upwards,to rather large values of ηrp. This shift is con-sistent with the qualitative phase diagram ofFig. 13 (right part). Of course, one must againrecall that in Fig. 16 we show “raw Monte Carlodata” for one choice of L only, and hence pro-nounced finite size tails near ηrp,cr(D) are appar-ent, as discussed for the case of capillary con-densation already (Fig. 6). The critical points

9

were again estimated from the cumulant inter-section method. Although strong corrections toscaling are present, the conclusion can be drawn[63] that the critical behavior of the interface lo-calization belongs to the d = 2 Ising universalityclass, as expected [36, 37, 38].

Figure 17 shows the phase diagram for D = 5and various choices of ε in the grand-canonicalrepresentation (this is the counterpart of Fig. 9afor capillary condensation, where D was variedfor ε = 0, while here we study the variation withε at fixed D). One can see that by increasingε the coexistence curves of the thin film movecloser towards the bulk coexistence curves, andfor ε = 2.5 the deviation from the bulk indeedis very small, but the critical point is stronglyshifted (from ηrp,cr = 0.766 in the bulk to ηrp,cr =1.106 in the thin film).

This behavior is qualitatively similar to whathas been found for the Ising ferromagnet withcompeting surface magnetic fields [11, 38], thegeneric model for which the interface localiza-tion transition was studied for the first time.

Note that the curve for ε = 0 in Fig. 17represents capillary condensation (and a simi-lar conclusion applies to the case ε = 0.5 aswell). Figure 17 implies that varying ε onecan completely smoothly cross over from cap-illary condensation-like behavior to interfacelocalization-like behavior, when ε is increased.In view of the qualitative description of Fig. 13this is somewhat surprising: in the capillarycondensation transition, the two liquid-vaporinterfaces bound to the walls annihilate eachother, the slit pore gets almost uniformly filledwith liquid. In the interface localization transi-tion, one has an interface on both sides of thetransition, it just has jumped at the transitionfrom one wall to the other.

How can one then reconcile Figs. 13 and 17with each other? The clue to the problem is, ofcourse, that the picture of the states in Fig. 13is far too simplified, it ignores the variationsof the densities close to the wall. Thereforethe states with “interfaces bound to walls” area simplification, which lose its meaning whenthe “phase” in between the interface and thewall can no longer be clearly identified withbulk-like properties (as is actually the case, see

Fig. 15). While one can clearly imagine to trans-form the left phase diagram of Fig. 13 into theright one by smooth changes, one should nottake the sketches that illustrate the characterof the phases too literally. The failure of thesesketches, however, also means that one must becareful with all approaches where wetting phe-nomena and interface localization transitions[24, 25, 26, 27, 36, 37, 38] are simply describedin terms of the “interface hamiltonian” picture,since according to this description the distanceℓ of the interface from the wall is the single de-gree of freedom (on a mean-field level) left toanalyze the problem.

5 Summary and Outlook

In this brief review we have emphasized thatconfinement has very interesting effects on softmatter systems, both with respect to the struc-ture and the phase behavior of these systems.Of course, confinement also has very interest-ing consequences on the dynamics of soft mat-ter systems (see e.g. [13, 14, 15, 16, 17, 18, 19]for recent discussions), but this aspect has beencompletely outside of the focus of our review.

We also have focused on the case which we con-sider to be the simplest case, confinement be-tween two flat and ideally parallel walls a fi-nite distance D apart. Practically more impor-tant, of course, is the confinement in randomporous media [10, 12, 20, 21, 22, 23]. However,in this case the random irregularity of the con-fining geometry is a serious obstacle for a de-tailed understanding. There is ample evidence(both from experiment [126, 127] and simula-tions [128, 129, 130]) that the liquid-vapor typephase separation or demixing of binary fluidmixtures under such confinement is seriouslymodified, but the character of this modifica-tion has been under discussion since a long time[126, 127, 128, 129, 130, 131, 132]. De Gennes[131] argued that due to the random arrange-ment of the pore walls (which prefer one of thecoexisting phases over the other) the problemcan be mapped to the random field Ising model[133, 134]. While for a long time the existing ev-idence [126, 127, 128, 129, 130] was inconsistent

10

with this suggestion, in recent work on colloid-polymer mixtures confined by a fraction of col-loids that are frozen in their positions and donot take part in the phase separation, evidencefor the random field Ising behavior was obtained[132]. Note that then it is necessary to reach aregime where the correlation length has grownto a large enough distance, much larger than thecharacteristic linear dimension of the confiningparticles.

Another case, that has received ample consider-ation in the literature (see [12] for further ref-erences) but was disregarded here, is the con-finement in a quasi-one-dimensional cylindri-cal geometry. In this case again the structureis typically inhomogeneous in the radial direc-tion perpendicular to the walls of the cylin-der. The correlation length can grow indefi-nitely only in one direction, along the cylinderaxis, however. Therefore the phase transitionto this laterally segregated state is a gradual(rounded) transition only, and even for condi-tions where in the bulk the system is stronglysegregated (with an interfacial tension γ be-tween the coexisting phases which is not small incomparison with kBT ), there is no macroscopicphase separation possible: rather one predictsthat only phase separation into domains of fi-nite size can occur (cross sectional area A of thecylinder and length ℓd of the domains), where[135, 136] ln ℓd ∝ Aγ/kBT . This happens be-cause thermal fluctuations prevent the estab-lishment of true long range order by the sponta-neous generation of transverse interfaces (acrossthe cylinder). This has the consequence thatalso, in principle, the capillary condensation orevaporation transitions in cylindrical geometryare, in full thermal equilibrium, not perfectlysharp but rounded. In practice, this effect oftenis masked by nonequilibrium phenomena (pro-nounced hysteresis occurs!) and hence we arenot aware of careful studies where this round-ing has been demonstrated. Note that mean-field treatments [12] miss such fluctuations ef-fects, of course. Also, when one considers cylin-drical geometries with “‘competing walls” (e.g.a cylinder with a square cross section, wherethe upper walls prefer one phase and the lowerwalls prefer the other phase [137, 138]) the in-

terface localization transition in such a “doublewedge”-geometry is rounded and not sharp, assimulations show where one considers the limitthat the length of the cylinder gets macroscopicwhile its cross section stays finite [138]. Sincerecently it has become possible to create artifi-cial cylindrical nanochannels with diameters be-tween 35 and 150 nm [139], it would be interest-ing to study phase separation in such nanochan-nels experimentally as well.

Using the AO model of colloid-polymer mix-tures as an example that is well suited for sim-ulation studies [61, 62, 63], we have discussedsimulation evidence for the theoretical conceptson capillary condensation and interface localiza-tion transitions [11, 33, 34, 35, 36, 37, 38]. Inparticular, the predictions for the shift of thecritical point have been found to be compati-ble with the simulation results, and it was alsoargued that the critical behavior of the lateralphase separation in the thin film has the char-acter of the two-dimensional Ising model (al-though in practice one is mostly in a crossoverregion where “effective exponents” in betweenthe d = 2 and d = 3 limits apply, which do nothave a deep theoretical significance). Clearly,it would be nice to have also experiments thatconfirm the findings of theory and simulation onthe phase behavior of confined fluid mixtures.

One crucial assumption of the work reviewedhere was that the wetting transitions (that oc-cur in the limit when the film thickness D di-verges to infinity) are of second order [11]. Iffirst-order wetting occurs, much more compli-cated phase diagrams under confinement result[11]. To some extent, this problem has beenworked out for symmetrical polymer blends un-der confinement [65, 66, 67, 68, 69, 70, 71, 72],and we refer the reader to these papers for de-tails. In particular, it also should be possibleto realize situations in between capillary con-densation and interface localization transitions[67].

Finally, we draw attention to more compli-cated ordering phenomena under confinement.A problem that has found a lot of attention isthe effect of confinement in thin films on theblockcopolymer mesophase ordering [140, 141,142, 143]. For more or less symmetric compo-

11

sition of a diblock copolymer, the mesophaseobserved in the bulk is a lamellar ordering [57].The question that is then discussed in the lit-erature (both experimentally and theoretically,see [140, 141, 142] for further references), iswhether the lamellae are oriented parallel orperpendicular to the confining walls, and tran-sitions in the number of lamellae that fit intothe thin film, etc. (remember that the lamellathickness depends on temperature, chain lengthof the polymers, and other control parameters[57]). For asymmetric compositions AfB1−f ofa diblock copolymer, however, already in thebulk melts other mesophases appear, such ashexagonal patterns of A-rich cylinders in B-richbackground, or cubic structures, where A-richcores of micelles form a periodic lattice in theB-rich background, or vice versa [57]. For tri-block copolymers, many much more complexmesophases occur, and the question how all thisself-assembly of block copolymers is affected byconfinement due to walls is still under boththeoretical and experimental investigation (seee.g. [143]).

Other very interesting confinement effects insoft matter occur when orientational order isinvolved, e.g. when a colloidal dispersion un-dergoing a transition from isotropic to nematicphases is confined by walls (see e.g. [144, 145,146, 147]). Confinement may enhance the ne-matic ordering tendency (“capillary nematiza-tion” [144] is the analog phenomenon of capil-lary condensation), but one needs also to takeinto account the tensor character of the orderparameter of liquid crystals. Thus near a walla biaxial character of the ordering occurs evenwhen in the bulk the ordering is uniaxial. Alsothe boundary conditions at the walls can be en-visaged such that one wall prefers parallel andthe other wall perpendicular alignment, lead-ing to a tilted structure of the ordering acrossthe film [145]. These remarks are by no meansintended as an exhaustive discussion, but justwant to draw the attention of the reader to thiswealth of interesting problems.

Acknowledgments: Support from the DeutscheForschungsgemeinschaft (DFG) via SFBTR6/A5 is gratefully acknowledged. We areindebted to S. Dietrich, R. Evans, H. Lekkerk-

erker, H. Lowen, M. Muller, M. Schmidt and P.Virnau for many stimulating discussions.

12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4

η p

ηc

0.5

0.7

0.9

1.1

1.3

0.0 0.1 0.2 0.3 0.4

ηr p

ηc

Figure 1: Phase diagram of the AO model witha size ratio q = Rp/Rc = 0.8 plotted in the(ηp, ηc) plane, showing the coexistence curvefor phase separation into a polymer-rich phase(left) and a colloid-rich phase (right), accordingto Monte Carlo (circles) and the free volumemean-field theory of Lekkerkerker et al. (fullline) [75]. The Monte Carlo data were obtainedusing a simulation box with linear dimensionsLx = Ly = 16.7, Lz = 33.4 (some data for asmaller box with Lx = Ly = 13.3, Lz = 26.5are also included as crosses). The solid squareshows the estimate for the critical point as ob-tained from a finite size scaling analysis of theMonte Carlo data, cf. text. The insert shows thesame data, in the so-called “reservoir represen-tation” where ηrp rather than ηp is used as a vari-able. The one-phase region where colloids andpolymers are fully miscible is always shown inthe lower parts of these diagrams, below the co-existence curves. From Vink and Horbach [81].

0

10

20

30

40

50

60

70

80

90

0.0 0.1 0.2 0.3 0.4

ln P

(ηc)

ηc

∆F

0.850.901.001.10

Figure 2: Logarithm of the probability P (ηc)of observing a colloid packing fraction ηc for anAO mixture with q = 0.8 at coexistence for sev-eral values of ηrp as indicated. The simulationswere performed in a box with linear dimensionsLx = Ly = 16.7 and Lz = 33.4 using periodicboundary conditions. Note that the distribu-tions are not normalized, and that for η nearηd = (ηVc + ηLc )/2 the distribution is essentiallyflat, almost independent from ηc, so that a freeenergy barrier ∆F is well-defined. From Vinkand Horbach [81].

13

0.0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4

γ*

ηcL − ηc

V

0.0

0.2

0.4

0.6

0.8

0.6 0.8 1.0 1.2

ηpr

Figure 3: Reduced interfacial tension γ∗ =4R2

cγLV/kBT for an AO mixture with q = 0.8as a function of the difference in packing frac-tion between the coexisting phase, ηLc −ηVc . Theinset shows γ∗ vs. ηrp. Open circles are the sim-ulation results, while the full curves are densityfunctional theory predictions. From Vink andHorbach [81]

0.75

0.76

0.77

0.78

0.79

0.80

0.00 0.05 0.10 0.15 0.20 0.25

η pr

ηc

two−phase region

L=21.0L=17.7L=16.7L=15.5

L=∞

Figure 4: Close-up of the phase diagram of theAO model for q = 0.8 in the critical region,showing only the range 0.75 ≤ ηrp ≤= 0.80 forthe polymer reservoir packing fraction. Resultsfor four different choices of the linear dimensionL of L × L × L simulation boxes are shown asbroken and dotted lines. The extrapolation to-wards the thermodynamic limit, L = ∞, also isincluded, and the estimate for the critical pointηrp,cr = 0.766± 0.002 is shown by the error bar.From Vink et al. [82]

14

−1.4

−1.3

−1.2

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

−1.0 0.0 1.0 2.0

ln(

y =

Mc

Lβ/ν )

ln( x = t L1/ν )

y = 0.27 xβ

L=21.0L=17.7L=16.7L=15.5

Figure 5: Scaling plot for the order parameterMc = 〈|m|〉 of the colloid-polymer mixture inthe bulk, for the AO model with q = 0.8, usingonly data for t > 0. The quantity Lβ/νMc isplotted vs. tL1/ν , choosing logarithmic scales.The straight line is a power law y = 0.27xβ,with β = 0.326. From Vink et al. [83].

0.0 0.1 0.2 0.3 0.4ηc

0.7

0.8

0.9

1.0

1.1

1.2

η prD=3.0D=5.0D=7.5D=10.0bulk

Figure 6: Coexistence curves of the AO modelwith q = 0.8 in the bulk (full curve withoutdata points, representing an extrapolation tothe thermodynamic limit) and for thin filmsconfined by symmetric walls, choosing Eq. (13)with ε = 0.5 as a wall potential. Open circlesdenote data for D = 3, L = 18 (all lengthshere are measured in units of the colloid diam-eter, σc = 2Rc). Open squares denote data forD = 5, L = 20; open diamonds are for D = 7.5,L = 30 , and open triangles for D = 10, L = 30.The broken curves denote the corresponding co-existence diameters, ηd = (ηLc + ηVc )/2 and thefull symbols denote estimates of the correspond-ing critical points for the various choices of D.From Vink et al. [62].

15

0.885 0.890 0.895 0.900ηp

r

0.6

0.7

0.8

0.9

U4

2.6 2.8 3.0 3.2

2.0

2.5

2d Ising

3d Ising

ln Y1

ln L

L=12.5

L=25.0

Figure 7: Moment ratio U4 for a film of thick-ness D = 5 plotted versus the polymer reser-voir packing fraction ηrp and several choices ofthe lateral linear dimension, L = 12.5, 15, 17.5,20, 22.5, and 25. The intersection points al-low to estimate the critical point as ηrp,cr =0.892± 0.002. The inset shows a log-log plot ofthe slope Y1 = dU4/dη

rp at ηrp,cr versus L, test-

ing the prediction that Y1 ∝ L1/ν . The straightline in the inset corresponds to an (effective)exponent νeff = 0.933. The broken horizontalstraight lines in the main part indicates the the-oretical values for the universal value U∗ of U4

at the intersection point, for both the d = 2 andthe d = 3 Ising model, respectively. From Vinket al. [62].

0.00

0.02

0.04

0.06

0.08

0.10

0.92 0.94 0.96 0.98

∆

ηpr

simulationfit

0.15

0.10

0.06

0.0410−110−210−3

t

∆

3D

2D

Figure 8: Order parameter ∆(= Mc) of the con-fined AO model with q = 0.8 plotted vs. ηrp,choosing D = 5 and ε = 0. The curve throughthe simulation data in the main frame is a fit to∆ = Befft

βeff , choosing ηrp,cr = 0.9223. The insetshows the same data as a function of the relativedistance from the critical point, t = ηrp/η

rp,cr−1,

on double logarithmic scales; broken straightlines illustrate the two-dimensional and threedimensional Ising exponents. From Vink etal. [61].

16

0.8 0.9 1.0 1.1ηp

r

3.0

4.0

5.0

µ ccoex

D = 3

D = 5

D = 7.5

D = 10

Bulk

0.0 0.1 0.2D

(∆1-∆)/ν

0.0

0.5

1.0

1.5

∆µc,

crco

ex(D

)

0.0 0.1 0.2D

-1/ν

0.0

0.1

0.2

0.3

∆ηp,

crr

(D)

a) b)

c)

Figure 9: a) Coexistence curves of the AOmodel with q = 0.8 in the grand-canonical rep-resentation where the chemical potential µcoex

of the colloids at the coexistence curve is plot-ted vs. the polymer reservoir packing fractionηrp. The bulk result (D = ∞) is shown as afull curve, while the broken curves show the co-existence curves for confined films for severalthicknesses D, as indicated. The symbols markthe corresponding critical points. b) Shift ofthe critical coexistence colloid chemical poten-tial plotted vs. D(∆1−∆)/ν . Equation (16) im-plies a straight line, as indicated by the dashedcurve. c) Shift of the critical polymer reservoirpacking fraction, ∆ηrp,cr(D), plotted vs. D−1/ν .The dashed lines indicate that Eq. (15) holds.From Vink et al. [62].

0.00 0.10 0.20 0.30D

-1

0.0

0.1

0.2

0.3

0.4

0.5

∆µ

ηpr = 1.2

ηpr = 1.1

ηpr = 1.0

Figure 10: Test of the Kelvin equation.The chemical potential difference ∆µcoex(D){Eq. (17)} is plotted vs. D−1 for three val-ues of ηrp, chosen well above the critical valuesηrp,cr(D). Broken straight lines show that thedata are compatible with the Kelvin equation.From Vink et al. [62].

Figure 11: Snapshot pictures of coexistingphases for the colloid-polymer mixture with q =0.8, D = 10, ηrp = 1.1, for ε = 0 (a) and ε = 2.0(b) Colloidal particles are in green, polymers inblue (the size of the polymers is rescaled to al-low a clearer view).

17

0 2 4 6 8 10z

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

η c(z)

ε = 0.0ε = 0.5ε = 2.0

0 2 4 6 8 10z

0.0

0.2

0.4

0.6

0.8

1.0

η p(z)

a) D=10, vapor

0 2 4 6 8 10z

0.0

0.5

1.0

η c(z)

ε = 0.0ε = 0.5ε = 2.0

0 2 4 6 8 10z

0.0

0.1

0.2

0.3

η p(z)

b) D=10, liquid

0 1 2 3z

0.0

0.1

0.2

0.3

0.4

η c(z)

ε = 0.0ε = 0.5ε = 2.0

0 1 2 3z

0.0

0.4

0.8

η p(z)

c) D=3, vapor

0 1 2 3z

0.0

0.5

1.0

1.5

η c(z) 0 1 2 3

z

0.0

0.1

0.2

η p(z)

d) D=3, liquid

Figure 12: Colloid density profiles obtainedin thin films at ηrp = 1.1, for two values ofthe film thickness D, and several values of thecolloid-wall parameter ε as indicated. Framesa) and b) show profiles obtained for D = 10,on the vapor and liquid branch of the coex-istence curve, respectively. Frames c) and d)show the corresponding profiles for thicknessD = 3. Note the jumps in the colloid density atz = 0.5, 1.0, 2.0, 2.5 caused through the jumpsof the potential at z = Rc and z = 2Rc, respec-tively. The insets represent density profiles ofthe polymers. From Vink et al. [62].

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D=

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0colloid−rich polymer−rich

asymmetric walls

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localizationtransition

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A II

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D finite

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B IIb

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B IIb

B IIaA I A II

Figure 13: Schematic phase diagram of acolloid-polymer mixture confined between twoparallel walls a distance D apart, in the grand-canonical ensemble where the polymer reservoirpacking fraction ηrp is used as ordinate and thedifference between the chemical potential of thecolloids at bulk phase coexistence µcoex(D = ∞)and the actual colloid chemical potential is usedas abscissa. Thus, phase coexistence in thebulk occurs along a vertical straight line atµcoex(∞)− µ = 0. The left part shows the caseof symmetric walls, the right part asymmetricwalls. States AI and AII coexist along the cap-illary condensation transition line, states BIIaand BIIb coexist along the interface localiza-tion line, while state BI exists along the brokencurve (which represents a line of rounded transi-tions). In the limit D → ∞, which correspondsto an infinite system but bounded by walls bothon the left and the right side, wetting transi-tions occur, that are rounded off for finite D.In the symmetric situation, the wetting transi-tions of both walls coincide at ηrp,w, while in theasymmetric situation ηr,rightp,w 6= ηr,leftp,w . From DeVirgiliis et al. [63].

18

Figure 14: a) Snapshot picture of the polymer-rich phase, for D = 10, q = 0.8, ηrp = 1.1. Thelower wall (at z = 0) has a wall potential param-eter energy ε0 = 0 and hence attracts colloidalparticles (shown in green), while the upper wall(at z = D) has a wall potential energy parame-ter εD = 2, attracting polymers (shown in blue).b) Snapshot picture of the same system as in a),but for ηrp = 0.9 showing a state with a delocal-ized interface.

0.0 2.0 4.0 6.0 8.0 10.0z

0.0

0.2

0.4

0.6

0.8

η c(z),

ηp(z

) ηc(z)ηp(z)

a) ε = 2.5, D = 10, ηc = 0.18, ηpr = 0.7

0.0 2.0 4.0 6.0 8.0 10.0z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

η c(z),

ηp(z

) ηc(z)ηp(z)

b) ε = 2.5, D = 10, ηc = 0.18, ηpr = 0.95

0.0 2.0 4.0 6.0 8.0 10.0z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

η c(z),

ηp(z

)

ηc(z)ηp(z)

c) ε = 2.5, D = 10, ηc = 0.05, ηpr = 1.20

0.0 2.0 4.0 6.0 8.0 10.0z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

η c(z),

ηp(z

) ηc(z)ηp(z)

d) ε = 2.5, D = 10, ηc = 0.33, ηpr = 1.20

Figure 15: Colloid concentration profiles ηc(z)and polymer concentration profiles ηp(z) as afunction of z for a thin film with asymmetricwalls (hard wall at z = 0, while for the otherwall at z = D = 10 the potential Ucw(h) acts seeEq. (13), with ε = 2.5). Profiles were obtainedat ηc = 0.18, ηrp = 0.70 (a) , ηc = 0.18, ηrp =0.95 (b), ηc = 0.05, ηrp = 1.20 (c), and ηc =0.33, ηrp = 1.20 (d). For profiles (c) and (d), thechoices ηc = 0.05, 0.33 roughly correspond tothe two branches of the coexistence curve, seeFig. 16. From De Virgiliis et al. [63].

19

0.0 0.1 0.2 0.3 0.4η

c

0.7

0.8

0.9

1.0

1.1

1.2

1.3

η pr

ε = 0.5ε = 1.5ε = 2.0ε = 2.5

a) D=5

bulk

L=30

0.0 0.1 0.2 0.3 0.4ηc

0.7

0.8

0.9

1.0

1.1

1.2

1.3

η pr

ε = 0.5ε = 1.5ε = 2.0ε = 2.5

b) D=10L=20

bulk

Figure 16: Coexistence curves for D = 5, L =30 (a) and D = 10, L = 20 (b), using fourvalues of ε, as indicated. Also the bulk coex-istence curve is shown (full curves). Full sym-bols mark critical points, the broken lines end-ing at these critical points are the coexistencediameters. The dotted horizontal straight linesmark the values of ηrp,cr(D). From De Virgiliiset al. [63].

0.9 1.0 1.1 1.2ηp

r

3.5

4.0

4.5

5.0

5.5

µ(D

,ε)

bulkε = 0.0ε = 0.5ε = 1.0ε = 1.5ε = 2.0ε = 2.5

Figure 17: Phase diagram of a thin film ofcolloid-polymer mixtures with asymmetric wallsin the grand-canonical ensemble, choosing q =0.8, D = 5, with ηrp as abscissa and µ as or-dinate. Curves show the coexistence potentialµcoex(D, ε) of the colloids. Full curve denotesthe result for the bulk (note that the bulk crit-ical point, ηrp,cr = 0.766 is off the scale of thisfigure). Full symbols mark the critical points ofthe films. From De Virgiliis et al. [63]

20

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