From capillary condensation to interface localization transitions in colloid-polymer mixtures confined in thin-film geometry.

Abstract

Monte Carlo simulations of the Asakura-Oosawa model for colloid-polymer mixtures confined between two parallel repulsive structureless walls are presented and analyzed in the light of current theories on capillary condensation and interface localization transitions. Choosing a polymer-to-colloid size ratio of q=0.8 and studying ultrathin films in the range of D=3 to D=10 colloid diameters thickness, grand canonical Monte Carlo methods are used; phase transitions are analyzed via finite size scaling, as in previous work on bulk systems and under confinement between identical types of walls. Unlike the latter work, inequivalent walls are used here: While the left wall has a hard-core repulsion for both polymers and colloids, at the right-hand wall an additional square-well repulsion of variable strength acting only on the colloids is present. We study how the phase separation into colloid-rich and colloid-poor phases occurring already in the bulk is modified by such a confinement. When the asymmetry of the wall-colloid interaction increases, the character of the transition smoothly changes from capillary condensation type to interface localization type. For very thin films (i.e., for D=3 ) and a suitable choice of the wall-colloid interactions, evidence is found that the critical behavior falls in the universality class of the two-dimensional Ising model. Otherwise, we observe crossover scaling between different universality classes (namely, the crossover from the three-dimensional to the two-dimensional Ising model universality class). The colloid and polymer density profiles across the film in the various phases are discussed, as well as the correlation of interfacial fluctuations in the direction parallel to the confining walls. The broadening of the interface between the coexisting colloid-rich and polymer-rich phases (located parallel to the confining walls) is understood in terms of capillary wave fluctuations. The experimental observability of all these phenomena is briefly discussed.

Full text

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arXiv:0807.4597v1 [cond-mat.soft] 29 Jul 2008
From Capillary Condensation to Interface Localization Transitions
in Colloid Polymer Mixtures Confined in Thin Film Geometry
Andres De Virgiliis1,2, Richard L. C. Vink3, J¨ urgen Horbach4, and Kurt Binder1
1Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Mainz,
Staudinger Weg 7, 55099 Mainz, Germany
2Instituto de Investigaciones Fisicoquimicas Teoricas y Aplicadas
(INIFTA), UNPL-CONICET, C.C. 16, Suc. 4, 1900 La Plata, Argentina
3Institut f¨ ur Theoretische Physik, Georg August-Universit¨ at G¨ ottingen,
Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany
4Institut f¨ ur Materialphysik im Weltraum, Deutsches Zentrum
f¨ ur Luft- und Raumfahrt (DLR), 51170 K¨ oln, Germany
Monte Carlo simulations of the Asakura-Oosawa (AO) model for colloid-polymer
mixtures confined between two parallel repulsive structureless walls are presented
and analyzed in the light of current theories on capillary condensation and interface
localization transitions. Choosing a polymer to colloid size ratio of q = 0.8 and
studying ultrathin films in the range of D = 3 to D = 10 colloid diameters thick-
ness, grand canonical Monte Carlo methods are used; phase transitions are analyzed
via finite size scaling, as in previous work on bulk systems and under confinement
between identical types of walls. Unlike the latter work, inequivalent walls are used
here: while the left wall has a hard-core repulsion for both polymers and colloids,
at the right wall an additional square-well repulsion of variable strength acting only
on the colloids is present. We study how the phase separation into colloid-rich and
colloid-poor phases occurring already in the bulk is modified by such a confine-
ment. When the asymmetry of the wall-colloid interaction increases, the character
of the transition smoothly changes from capillary condensation-type to interface
localization-type. The critical behavior of these transitions is discussed, as well as
the colloid and polymer density profiles across the film in the various phases, and the
correlation of interfacial fluctuations in the direction parallel to the confining walls.
The experimental observability of these phenomena also is briefly discussed.

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I.INTRODUCTION AND OVERVIEW
When fluid systems are confined in nanoscopic pores or channels, one expects that the
phase behavior can be profoundly modified [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Such effects
have found an increasing attention recently, for instance because of the current interest
to fabricate devices of nanoscopic size and to manipulate chemical reactions in nanoscopic
reaction volumes (“lab on a chip”), etc. [11, 12, 13, 14, 15, 16]. In addition, porous materials
with pores of nanoscopic widths are useful as catalysts or for applications such as mixture
separation, pollution control, etc. [6, 17, 18, 19].
However, such applications often are based on empirical knowledge, the theoretical un-
derstanding of confined fluids still being rather limited [6, 7, 8, 9, 10]. In order to make
progress with the theoretical description of fluids under confinement by the methods of
statistical thermodynamics, it is desirable to start with relatively simple model systems,
where both the geometry of confinement is well characterized, and the relevant interactions
among the fluid particles and between the fluid particles and the confining solid surfaces
are sufficiently well understood. Last but not least, suitable experimental tools should be in
principle available to put the theoretical predictions to a test.
For these purposes it is hence useful to consider colloidal suspensions [20, 21, 22, 23,
24], exploiting the analogy between colloidal fluids and fluids formed from small molecules,
but taking advantage of the much larger length scales (in the µm range), of the colloidal
particles. Such systems allow detailed experiments in which individual particles can be
tracked through space in real time using confocal microscopy techniques [25]. Particularly
useful systems in the present context are colloid-polymer mixtures, which can undergo in
the bulk a liquid-vapor like phase separation into a colloid-rich phase (the “liquid”) and a
colloid-poor phase (the “vapor”) [23, 26]. This phase separation is due to the (entropic)
depletion attraction between the colloids caused by the polymers. A very simple model,
due to Asakura and Oosawa [27] and Vrij [28] describes the resulting phase separation in
the bulk [29, 30, 31, 32, 33] in excellent qualitative agreement with the experiment [23].
While initially it was thought that mean field theory [29] accounts very accurately for the
Monte Carlo (MC) simulation results [30, 31] of this Asakura-Oosawa (AO) model, a more
extensive MC simulation study [32, 33] revealed clear evidence for Ising-like critical behavior
[34] over a broad regime of control parameters.

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When such a colloid-polymer mixture is confined by hard walls, also a depletion attraction
of the colloids and the walls occurs [35] and can cause (in semi-infinite geometry [36, 37,
38, 39, 40]) the formation of wetting layers [41, 42, 43, 44, 45, 46]. Due to the very low
interfacial tension between unmixed phases [47, 48, 49, 50], thermally activated capillary-
wave fluctuations [51, 52, 53, 54, 55] are readily observable in experiment [56] and simulation
[50]. The phase behavior of colloid-polymer mixtures in confinement can be also studied
experimentally. Therefore, this issue has been addressed in recent computer simulation
studies, considering the confinement of colloid-polymer mixtures by two parallel hard walls
a distance D apart [9, 57, 58, 59]. These studies have confirmed the fact that lateral phase
separation in a thin film geometry exhibits a critical behavior belonging to the class of the
two-dimensional Ising model [58]. Also the the scaling relations of Fisher and Nakanishi [60]
have been verified. Unlike the case of confinement of small molecule fluids in nanopores, the
size of the particles in colloidal fluids by far exceeds the scale of the atomistic corrugation
of the pore walls, and hence the effects of this corrugation on the packing of particles near
the walls [61, 62] need not be considered here.
A very useful aspect of colloidal suspensions is that interactions among such particles can
be tuned by suitable surface treatment [20, 21, 22, 63]. E.g., a short-range repulsion between
colloidal particles often is created by coating them with a polymer brush [63, 64]. Similarly,
one could cancel (partially or completely) the depletion attraction of colloids towards a hard
wall by coating the latter with a polymer brush, choosing the grafting density and chain
length of these flexible polymers appropriately.In a colloid-polymer mixture, however,
for moderate chain stretching in the polymer brush the polymers in the solution still can
penetrate into the brush, experiencing hence a much weaker interaction than the colloidal
particles. Only for strongly stretched chains, as occurring in very dense polymer brushes
[65], a repulsion of the polymer coils in the solution would result as well, even if the chemical
nature of the polymers in the solution and in the brush is identical (“autophobicity effect”
[66, 67]).
This tunability of the wall-colloid interactions opens the possibility to realize a situation
of a slit pore with asymmetric walls: suppose the left wall is simply a hard wall, attractive for
the colloids, and the right wall a coated hard wall, repulsive for the colloids (Fig. 1) [68]. With
a colloid-polymer mixture confined between such asymmetric walls, the possibility arises to
realize the “interface localization transition” [7, 9, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. This

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transition is illustrated in Fig. 1. Here, the so-called “polymer reservoir packing fraction”
is defined by ηr
p≡ (4π/3)R3
pexp(µp/kBT) (with Rp and µp the radius and the chemical
potential of the polymers, respectively) and plays the role of inverse temperature when
we compare the behavior to that of a fluid of small molecules that undergoes a liquid-
vapor transition. While in the bulk colloid-polymer mixture phase separation sets in when
the variable ηr
pexceeds the critical value ηr
p,crit, this transition is rounded in the thin film.
Starting out from a layer enriched with colloids on the left wall and enriched with polymers
at the right wall, a stratified domain structure forms, with a domain wall separating the
colloid-rich phase in the left part and the polymer-rich phase in the right part of the slit
pore (state BI in Fig. 1). Only at a much larger value ηr
p,crit(D) a sharp phase transition
occurs in the thin film, with the colloid-polymer interface being bound either to the right
wall (phase B IIa) or to the left wall (phase B IIb). Along the line µ = µcoex(D,ηr
p) these
two phases may coexist.
Of course, in an experiment one does not have at one’s disposal the intensive variables µ
and the “polymer reservoir packing fraction” ηr
p, but rather the volume fractions of colloids
and polymers,
ηc=4π
3R3
cNc/V ,ηp=4π
3R3
pNp/V ,(1)
where V is the volume of the system, Rcthe radius of the spherical colloidal particles, and
Nc, Np are the particle numbers of colloids and polymers, respectively. Since ηc, ηp are
densities of extensive thermodynamic variables, the first order transition lines µcoex(D,ηr
p)
in the plane of variables ηc, ηpare split into two phase coexistence regions. Bringing the
thin film from the one-phase region to inside the two-phase region (e.g. by adding polymers
to the solution), one creates a state of the slit pore where in parts of the system the interface
is bound to the left wall and in other parts it is bound to the right wall. These phases are
then separated by interfaces running across the film from the left to the right wall (or vice
versa). A similar phase coexistence between the two phases AI, AII occurs in the case of
capillary condensation-like transitions for symmetric walls (left part of Fig. 1). As always,
the amounts of the coexisting phases is controlled by the lever rule.
In the limit D → ∞ of the film thickness, we recover a semi-infinite system and then
wetting transitions are expected to occur, so that, in the symmetric wall case, in the region
ηr
p,crit< ηr
p< ηr
p,wfor µ = µcoex(∞) both walls are (completely) wet, while for ηr
the walls are nonwet (“incomplete wetting” [36, 37, 38, 39, 40]). In fact, the colloid-rich
p> ηr
p,w

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surface enrichment layers indicated for the phase AII are the precursors of wetting layers
that appear when D → ∞. Of course, no (infinitely thick [36, 37, 38, 39, 40]) true wetting
layer fits into a thin film of finite thickness D, and thus the wetting transition at ηr
p= ηr
p,w
(which we have assumed to be of second order [36, 37, 38, 39, 40]) is rounded off in the thin
film.
For asymmetric walls in the limit D → ∞ the wetting transitions at both walls will occur,
in general, for different values of ηr
pat both walls. In Fig. 1 we have arbitrarily assumed
that ηr,left
p,w> ηr,right
p,w
. In the simplistic Ising model with “competing surface magnetic fields”
[69, 70, 71, 72, 73, 74] H1and HD, one can consider a situation with HD= −H1, where
these transitions then coincide, ηr,left
p,wp,w = ηr,right
. However, such a special symmetry never is
expected for a colloid-polymer mixture (which has an asymmetric phase diagram already
in the bulk).Note, however, that for D → ∞ one does not expect that for interface
localization transitions ηr
p,crit(D) converges to the bulk critical point, ηr
p,crit: rather one
expects a convergence towards the wetting transition which is closest to the bulk transition
[7].
In the present paper, we shall present evidence from Monte Carlo simulations that the
scenario sketched in Fig. 1 is correct, and we shall characterize the behavior of colloid-
polymer mixtures confined by asymmetric walls in detail, considerably extending preliminary
work [68]. Extensive results for the case of symmetric walls have been presented earlier [58,
59]. As in previous studies in the bulk [32, 33] the simulations are carried out mostly in the
grand-canonical ensemble, using a dedicated grand-canonical cluster algorithm [32] together
with re-weighting schemes such as successive umbrella sampling [79]. Phase transitions
are analyzed by finite size scaling methods [80, 81, 82], varying suitably the lateral linear
dimensions L along the walls. For a description of these techniques, the reader should consult
our earlier work [58, 59].
In Sec. II we now present a study of the “soft mode” phase [72] BI for a relatively thick
film (thickness D = 10 colloid diameters). Such phases with delocalized interfaces are of
great interest due to their large interfacial fluctuations [72, 73, 74, 83, 84], and consequences
of such fluctuations have been seen in experiments both on polymer blends [85] and colloid-
polymer mixtures [46]. Sec. III then gives a discussion of the interface localization transition
for an ultrathin film (D = 3), attempting to verify the above statement that the critical
exponents should be those of the two-dimensional Ising model. Sec. IV discusses the phase

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behavior when both film thickness and the strength of the short range colloid-wall repulsion
are varied. Finally, Sec. V summarizes some conclusions.
II.FORMATION AND PROPERTIES OF THE INTERFACE IN THE SOFT
MODE PHASE
All our Monte Carlo simulations refer to the standard Asakura-Oosawa (AO) model and
use the same size ratio q = Rp/Rc= 0.8 as the previous work in the bulk [32, 33] and for
symmetric walls [58, 59]. In this case, it is known that the critical point in the bulk occurs
at [32, 33]
ηr
p,crit= 0.766 ± 0.002,ηc,crit= 0.1340 ± 0.0002,ηp,crit= 0.3562 ± 0.0006,(2)
and also the coexistence curve between the colloid-rich phase (ηc,ℓ) and the polymer-rich
phase (ηc,v) is known rather precisely, as well as the interfacial tension [32, 33, 50]. We
now consider a L × L × D geometry, where all lengths are measured in units of the colloid
diameter 2Rc, and periodic boundary conditions are applied in x and y-directions only. For
the thickness D, the values D = 3, 5, 7, and 10 are used, while the linear dimension L in
parallel direction is chosen in the range from L = 15 to L = 30. The left wall, located
at z = 0, is taken purely repulsive for both colloids and polymers. As for the interaction
between the colloidal particles (which is infinite if two colloids overlap and zero else, as
well as the colloid-polymer interaction which also is infinite if a colloid particle overlaps a
polymer and zero else), we take a hard wall repulsion,
Uℓ
w,c(z) = ∞, z < Rc,
Uℓ
Uℓ
w,c(z) = 0, z > Rc,(3)
w,p(z) = ∞, z < Rp,Uℓ
w,p(z) = 0, z > Rp, (4)
for both colloids [Uℓ
w,c(z)] and polymers [Uℓ
w,p(z)]. At the right wall, however, we add a
square well potential of strength ε and with an additional range Rc. Thus, the potential
acting on the colloids is
Ur
w,c(z) = 0,z < D − 2Rc,
D − 2Rc< z < D − Rc,
z > D − Rc.
(5a)
Ur
w,c(z) = ε, (5b)
Ur
w,c(z) = ∞, (5c)

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This square well potential [Eq. (5b)] could be realized by a polymer brush of low grafting
density and height Rc, for instance, so that the region of z where the colloid penetrates into
the brush leads to a finite energy penalty ε only (note that we use the convention that the
temperature kBT = 1; of course, one could also consider square well potentials of arbitrary
range). For the polymers, on the other hand, the interaction is taken to be of the same type
as in Eq. (4),
Ur
w,p(z) = 0, z < D − Rp,Ur
w,p(z) = ∞, z > D − Rp.(6)
This potential models the interactions of polymers with a hard wall coated with polymer
brushes: Under good solvent or Theta solvent conditions [86], polymers can overlap with
weakly stretched polymer brushes with little free energy cost.
It turns out that a phase behavior as sketched in the right part of Fig. 1 occurs if ε ≥ 2.5.
Figure 2 presents some typical profiles of the average local volume fraction of colloids ηc(z)
and polymers ηp(z) across the slit pore, for the case ε = 2.5 and D = 10. Panel (a) shows the
profiles for ηc= 0.18 and ηr
p= 0.7, corresponding to a state point where the bulk colloid-
polymer mixture is still in the one-phase region. Nevertheless, the profiles of ηc(z) and
ηp(z) exhibit pronounced inhomogeneities: the polymer profile ηc(z) displays a pronounced
peak close to the right wall, and decays with increasing distance from the right wall to a
plateau, almost independent of z, in the regime 3 ≤ z ≤ 6. Very close to the left wall,
where the volume fraction of colloids is strongly enhanced, the concentration of polymers
is also inhomogeneous (indirectly induced by the colloids, since polymers and colloids must
not overlap), before ηp(z) abruptly decreases to zero for z = Rp. The colloid profile ηc(z)
shows a very pronounced peak close to z = Rc, on the other hand, which can be attributed
to the depletion attraction of the colloids to the hard wall. One can recognize a second peak
near z = 1.6 and a weak third peak near z = 2.5, these peaks represent the well-known
“layering” of hard particles near smooth repulsive walls. In the central part of the thin film,
for 3 ≤ z ≤ 6, the profile ηc(z) is almost flat; thus the surface enrichment of the colloidal
particles at the hard wall is a short range effect. In the regime near the right walls, where the
polymers are attracted, we recognize first a smooth decrease of ηc(z) in the range where the
pronounced increase of ηp(z) sets in. For z = D − 2Rc= 9, where the additional repulsive
potential sets in, a downward step in ηc(z) occurs, as expected.
It is interesting to contrast the behavior in panel (a), showing surface enrichment of
colloids (left) and polymers (right) at the walls confining an otherwise homogeneous mixture,

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with the behavior in panel (b), which refers to a state where in the bulk phase separation
has occurred. Indeed, Fig. 2b gives rather clear evidence for a phase separation in the z-
direction perpendicular to the confining walls, of the type denoted as BI in Fig. 1. The
polymer rich phase occurs on the right side of the thin film, and ηp(z) reaches very small
values for z ≤ 4. Near z = 6 we recognize inflection points in both profiles ηp(z), ηc(z) as are
typical for interfaces between coexisting phases. Again the profile ηc(z) exhibits the typical
layering oscillations for small z. No such layering occurs for the polymers near z = D, of
course, since the polymer-polymer interaction is zero, the polymer-rich phase is like a dense
ideal gas.
Panels 2c and 2d illustrate states corresponding to the phases BIIb and BIIa in Fig. 1,
respectively. In the polymer-rich phase the interface position is at about z = 2.5, and unlike
Fig. 2a (where the interface is freely fluctuating in the center of the slit pore) the width of
the interface is only about two colloid diameters. Such a state is typical for a colloid-polymer
interface tightly bound to the left wall. Figure 2d is the counterpart showing the profiles
in the colloid-rich phase, where almost all polymers are expelled, apart from the immediate
neighborhood of the right wall.
We conclude that these profiles do give qualitative evidence for the existence of all three
phases BI, BIIa and BIIb in Fig. 1. We now study the phase with the delocalized interface
(BI) more closely. In particular, we are interested in how the interfacial profiles change when
the inverse-temperature-like variable ηr
pis varied (Fig. 3). Defining an order parameter m
and the coexistence diameter δ as follows,
m = (ηℓ
c− ηv
c)/2,δ = (ηv
c+ ηℓ
c)/2, (7)
we choose the average volume fraction of the colloids such that ηc= δ, and we attempt to
fit the colloid density profile by a tanh function,
ηc(z) = δ − mtanh[(z − z0)/w]; (8)
here z0is the position of the interface center and w is the interfacial width. Fig. 3a shows
that Eq. (8) provides a good fit of the colloid density profile, for all values of ηr
pfrom 0.90
to 1.10. For ηr
p= 0.80, however, the profile is extremely wide, due to the proximity of the
critical point in the bulk [Eq. (1)], and then the fit is less convincing. Indeed, the polymer
density profile ηp(z), Fig. 3b, for ηr
p= 0.8 does not even exhibit an inflection point, while

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for all larger values of ηr
pan inflection point clearly is present (it occurs roughly at z = z0,
the inflection point of the polymer density profile, which is roughly at z0≈ 0.20 ± 0.05).
Of course, one notes that ηc(z) does not reach the regime of homogeneous “liquid” density
ηℓ
c, since for z ≤ −1.5 in Fig. 3a the layering effect caused by the repulsive wall at z = −5
already sets in. Likewise, the surface enrichment of the polymers at the right wall distorts
the profiles for z ≥ 3.5 in Fig. 3b. We also note that the profiles seem to have common
intersection points (which do not coincide with z0, since both m and δ depend on ηr
p). The
common intersection point of the colloid profiles is at z = 0.5 ± 0.05, while the common
intersection point of the polymer profile is at z = 0.0 ± 0.1. Presumably, these common
intersection points are just numerical coincidences, and will not occur in the general case
(using other choices of ε and D, for instance). However, the statistical effort for the data
in Fig. 3 is rather substantial, and hence no such systematic parameter variation has been
attempted.
Figure 3c shows that the effective interfacial width w extracted from the fit to Eq. (8)
increases from about w ≈ 1.5 near the critical point of the thin film (the estimation of thin
film critical points is discussed in the following sections) to about w ≈ 2.4 for ηr
However, it is important to recall that the width w of the interface in the “soft mode”
p= 0.8.
phase depends on both ηr
pand the total film thickness D [83, 84, 85, 87]. This complicated
behavior results because the “intrinsic interfacial profile” [88, 89] is broadened by capillary
waves [51, 52, 53, 54, 55], but the long-wavelength part of the capillary wave spectrum is
suppressed by the effective interface potential [38, 39] caused by the walls. For short range
forces due to the walls, as occurring here, the corresponding prediction for the mean square
width is [83, 84, 85, 87]
w2= w2
0[1 +ωπ/4
2 + ω
D
w0] + const (9)
Here, w0 is the “intrinsic width”, which should be related to the correlation length ξb
along the coexistence curve in the critical region, w0= 2ξb, while the wetting parameter
ω [38, 39, 40, 90, 91, 92, 93] for Ising-like systems is ω ≈ 0.8 and the (unknown) constant
due to the short wavelength cutoff needed in the capillary wave spectrum [83, 84, 85] can
be neglected near the critical point of the bulk. The intrinsic width should then vary with
ηr
pas
w0= ˆ w0(ηr
p/ηr
p,crit− 1)−ν,ν ≈ 0.63,(10)

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with an amplitude factor ˆ w0which is presumably in the range 0.2 ≤ ˆ w0 ≤ 0.5 (it is not
accurately known since an unambiguous separation of intrinsic width and capillary wave
broadening is hardly possible in interfacial profiles [50, 87]). Since for the chosen values of
pwe have D ≫ w0for D = 10 and (ωπ/4)/(2+ω) ≈ 0.224, we expect that w ≈ 1.497√w0
in our case, i.e. w in Fig. 3c should increase with an exponent ν/2. Disregarding the results
ηr
for ηr
p= 0.8 and ηr
p= 0.85, which are too close to ηr
p,critand hence unreliable due to finite
size effects, we find that the remaining data for L = 120 can be nicely fitted to a critical
power law with the expected exponent ν/2 = 0.315 (see insert of Fig. 3c).
Thus, it clearly would be of interest to obtain reliable data close to the bulk critical point,
but then much larger systems would be required, and this would require very substantial
computer resources, that are not available to us. But we emphasize the fact that no singular
behavior can be observed when at fixed D we vary ηr
pthroughout the bulk critical region,
passing the critical point. As an example, Fig. 4 shows density profiles for the case D = 10,
ηc= 0.195,L = 40 and three values of ηr
pclose to ηr
p,crit[Eq. (1)]. One sees that profiles for
ηr
pslightly above ηr
p,critand slightly below it are hardly distinct from each other, all changes
with respect to ηr
pare very gradual.
A very interesting property is the correlation function of the colloidal particles in the
interfacial region, z0− w < z < z0+ w (see Fig. 5). If we were to consider an unconfined
interface, the capillary wave fluctuations would cause a power law decay of these fluctuations.
Due to the confinement, the interface feels an effective potential, and this leads to the
existence of a finite correlation length ξ||of interfacial fluctuations, as discussed extensively
in the literature [72, 74, 83, 84, 85, 87]. In simulations of a model for a symmetrical polymer
mixture confined between competing walls, this correlation length was studied as a function
of film thickness. Here we rather study this quantity as the interface localization transition
is approached. Figure 5a shows that the radial distribution function of colloidal particles in
the interfacial regions is well described by the formula
gc(r) = const exp(−r/ξ||)/√r.(11)
Equation (11) was also shown to work very well in the case of the symmetric polymer
mixture [83]. When ηr
papproaches the value ηr
p,crit(D), one sees a strong increase of ξ||,
reflecting the expected critical divergence of ξ||at the interface localization transition [which
occurs at about ηr
p,crit(D) ≈ 1.13±0.03]. Arguments have been given to show that for large

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enough D there is a region of mean-field like behavior, where ξ||∝ (1 − ηr
with ν||= 1/2, while very close to ηr
p/ηr
p,crit(D))−ν||
p,crit(D) the critical behavior should fall in the class of
the two-dimensional Ising model [74], ν||= 1. However, the accuracy of the data in Fig. 5b
does not warrant an analysis of this crossover behavior.
III.INTERFACE LOCALIZATION TRANSITION IN VERY THIN FILMS
Following the procedures used in our earlier study of capillary condensation in the AO
model, we carried out a finite size scaling analysis of the model with ε = 3.0 for a slit pore
which is only D = 3 colloid diameters thick. Varying the chemical potential and applying
successive umbrella sampling [79], the probability distribution P(ηc) is recorded. Applying
suitable re-weighting techniques [94], one can apply the equal area rule [95, 96] to determine
the chemical potential µcoexwhere the peak of P(ηc) representing the vapor-like phase and
the peak representing the liquid-like phase have equal weight. Figure 6a shows typical data
near the second order interface localization transition of the thin film, and Fig. 6b shows
the fourth order cumulant U4 as a function of ηr
pfor various L from L = 15 to L = 30.
Introducing an order parameter M as M = ηc− ?ηc?, the moments ?Mk? are defined as
?Mk? =
1
?
0
MkP(ηc)dηc,(12)
and U4then is given as the ratio of the square of the second moment and the fourth moment,
U4= ?M2?2/?M4? .(13)
For large enough L, when finite size scaling [80, 81, 82] holds, a convenient recipe to find
the critical point ηr
p,critis to record U4for different choices of L versus ηr
ptuning µ such that
µ = µcoex(ηr
p) and look for a common intersection point [80]. For ηr
p≤ ηr
p,critone fixes µ by
the criterion that ?M2? is maximal {for ηr
estimate µcoex(ηr
p> ηr
p,critthis criterion is an alternative way to
p)}.
Figure 6a indicates the gradual change from a double peak distribution to a single peak
distribution, which is a characteristic behavior for all second order phase transitions. Note
that ηr
p,critdoes not correspond to the value of ηr
pwhere P(ηc) becomes flat over a broad range
of ηc: rather ηr
p,crit(D) still corresponds to a double peak distribution [80, 81, 82]. Figure

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6b yields ηr
p,crit(D = 3) = 1.300 ± 0.005, i.e. a value very far away from ηr
[cf. Eq. (1)]. Although it is somewhat disappointing that one cannot really find a unique
p,critin the bulk
intersection point of the cumulants U4(ηr
p) for the various choices of L, one must recognize
that for high enough resolution of the coordinate axes such a scatter is quite expected, due
to residual corrections to finite size scaling [80], and due to the statistical errors of the
Monte Carlo data [97]. More disturbing is the fact that the cumulant intersections occur in
a range of values in between the universal constants U∗(2dim) and U∗(3dim) for the two-
and three-dimensional Ising model [98, 99], respectively,
U∗(2dim) ≈ 0.856,U∗(3dim) = 0.629 .(14)
As Fig. 6b shows, intersections occur in the range 0.73 < U∗< 0.80 (although there is some
tendency of the intersection points to move upward with increasing L). On the other hand,
the slope of the cumulants at the intersection point, which is predicted to scale as [80]
dU4/dηr
p∝ L1/ν,(15)
yields an effective exponent rather close to the prediction ν = 1 for the two-dimensional
Ising model.
Figure 7a shows simulation results for the order parameter m = (ηℓ
c− ηv
c)/2, where the
volume fractions of colloids ηℓ
c, ηv
care not read off from the peak positions in Fig. 6a, since
for shallow peaks this would be a somewhat arbitrary procedure, but rather we take m as
the first moment of the absolute value ?|M|?. Similarly, Fig. 7b shows the “susceptibility”
χ0= L2D(?M2? − ?|M|?2). Both quantities are very strongly affected by finite size effects:
Rather than exhibiting a power law decay, m ∝ (1−ηr
approaching ηr
p,crit/ηr
p)βwith β = 1/8, one finds that
p,critfrom above, the curves for m for the different values of L splay out and
develop very pronounced “finite size tails” [80, 95] for ηr
p< ηr
p,crit. At η = ηr
p,critone finds
that the data are compatible with a power law decay (insert to Fig. 7a)
m∗(L) ≡ m(L,ηr
p,crit) ∝ L−β/ν.(16)
According to the two-dimensional Ising model, one would expect β/ν = 1/8. However, the
straight line in the insert of Fig. 7a rather indicates an effective exponent of (β/ν)eff ≈
0.20 ± 0.02. Likewise, the susceptibility maxima, which should scale as [80, 81, 82]
χmax
c
∝ Lγ/ν, (17)

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with the two-dimensional Ising value being γ/ν = 1.75, rather suggest an effective exponent
(γ/ν)eff= 1.60 ± 0.03. Very roughly, these exponents are compatible with the hyperscaling
relation [34] γ/ν + 2β/ν = 2. Using the quoted effective exponents 1/νeff, (β/ν)eff and
(γ/ν)eff, one finds that on a scaling plot, where the variable t ≡ |ηr
with L1/νand m or χ are rescaled with Lβ/νor L−γ/ν, one finds reasonable data collapse
p,crit/ηr
p− 1| is rescaled
(Fig. 8). Such a partial success of a finite size scaling analysis, i.e. good data collapse is only
found when effective exponents are used that deviate somewhat from the theoretical values,
has already been seen for interface localization-delocalization transitions in the Ising model
[73, 74] and hence these problems are not a surprise in the present case.
IV.OVERVIEW OF THE PHASE BEHAVIOR
We now describe some of our results for other film thicknesses D. In principle, the same
type of analysis was carried out for D = 5 and D = 7 as well, but it turned out that
the distribution P(ηc) for ηc > ηc,crit(D) becomes increasingly asymmetric when D gets
larger (Fig. 9). Also the cumulant intersections get spread out over a rather large range
of ηr
p(Fig. 10), and these intersection points lie even in a range that is below the three-
dimensional Ising value, Eq. (14). We interpret this finding as an indication that with D
getting larger an increasing fraction of the critical region falls into the region of mean-field
like behavior, as was theoretically predicted [74].
Also for fixed D the accuracy, with which ηr
p,crit(D) can be estimated, clearly deteriorates
when ε increases (Fig. 11). Note that data for D = 5 and ε = 1.0 were already given in our
preliminary communication [68], the choice ε = 1.0 corresponds to a capillary condensation-
type behavior, however.
Figure 12a shows estimates for the phase diagrams for the interface localization transition
for ε = 3 and three choices of D, while Fig. 12b shows analogous data for D = 5 but varying
ε, and Fig. 13 shows a plot of ηr
p,crit(D = 5) vs. ε. One sees that miscibility is enhanced if
either D decreases, or ε increases, or both.
Finally we turn to the variation of ηr
p,critwith ε for the choice D = 5 (Fig. 13). As
found from a self-consistent field calculation for a symmetrical polymer mixture confined
between competing walls [77], the minimum of the curve ηr
p,critdoes not occur for the case
of symmetric walls (ε = 0), but for an asymmetric situation. It also is remarkable and

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unexpected, that for large ε the curve for ηr
p,critdoes not level off.
Figure 14 shows the counterpart of the schematic Fig. 1 (left part), presenting in the
plane of variables µc and ηr
p(D) the numerical results for the coexistence curves between
colloid-rich and polymer-rich phases, for the case of 2.0 ≤ ε ≤ 4.0, i.e. the region where
interface-localization transitions occur (which are highlighted in the diagram by arrows).
Note that unlike Fig. 1, µc(∞) was not subtracted from µc, thus the bulk coexistence is not
simply the ordinate axis as in Fig. 1, but rather a nontrivial curve (which actually is not very
different from a straight line). While for ε = 2.0 there is still a small but systematic offset
between the curves µc(ηr
p,D = 5) and µc(∞), for ε = 3.0 and ε = 4.0 the offset is almost
negligibly small. The part of the curves µcoex(ηr
p(D)) to the left of ηr
p,crit(D) represents the
state BI in the schematic phase diagram, Fig. 1, where a delocalized interface occurs in
the center of the film, separating the colloid-rich phase adjacent to the left wall and the
polymer-rich phase adjacent to the right wall.
At this point, we return to the density profiles at phase coexistence, and compare them
for the same choice of ηr
pand D = 10, but different values of ε, ε = 2.0 and ε = 4.0 (Fig. 15).
For ηr
p= 1.5, the vapor-like phase reaches the same polymer density for both choices of ε;
the main difference concerns the colloid-rich side of the systems, the colloid enrichment at
the hard wall is more pronounced for ε = 4.0 than for ε = 2.0. However, in the liquid-
like colloid-rich phase the behavior is just the other way round: the layered profiles of the
colloid-rich phase near the hard all are virtually identical, while the polymer enrichment
near the right wall is more pronounced for ε = 4.0 than for ε = 2.0. When one studies the
effect of varying ε in the one phase region for ηr
p< ηr
p,crit(D) however, one sees only a minor
effect of ε on the segregated structure where an interface has formed parallel to the walls
(Fig. 15b and 15d), in particular for not extremely thin films.
V.CONCLUSIONS
In this paper, the Asakura-Oosawa (AO) model for colloid-polymer mixtures for a size
ratio of polymers to colloids q = 0.8 was studied by Monte Carlo simulation, considering thin
films of thickness D = 3 to D = 10 colloid diameters and confinement between asymmetric
walls. One wall is simply a repulsive hard wall, to which the colloidal particles are attracted
via depletion forces; the other wall exerts a square-well-type repulsive interaction (of the

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range of the colloid ratio, and variable strength ε = 0.5 to 4.0, in units of kBT = 1.0). This
study complements our earlier work on the AO model in the bulk, and under confinement
between two equivalent hard walls, where capillary-condensation like phenomena occur; for
the present model, we can smoothly interpolate from capillary condensation-like behavior
for small ε (e.g. ε = 0.5 or 1.0), when both walls show some (though unequal) surface
enrichment of colloids, to interface localization-type transitions, occurring for large ε (e.g. for
ε varying from ε = 2.5 to ε = 4.0). In the latter case, only the hard wall attracts colloids
while the other wall attracts polymers. In this region, for large D the precise value of ε
has little effect on the observed density profiles. When one then increases the polymer
reservoir packing fraction ηr
p(which plays an analogous role as the inverse temperature does
for thermally driven phase separation in small molecules mixtures), one observes that the
enrichment layers of colloids and polymers at the walls gradually transform into two domains
of coexisting colloid-rich and polymer-rich phases, separated by an interface parallel to the
confining walls. We find that the temperature dependence of the width of this interface
is considerably weaker than that of the bulk correlation length (or “intrinsic” interfacial
width, respectively), and account for this finding in terms of capillary wave broadening of
the interface. However, since for D < 10 the interface profiles are strongly affected by
layering of colloids near the hard wall, study of this broadening is difficult.
Only far away from the bulk critical point can a sharp phase transition be observed, which
we analyze by finite size scaling methods. While for D = 3 and not too large ε the critical
value ηr
p,crit(D) can be rather accurately determined, and evidence can be found that the
critical behavior falls in the universality class of the two-dimensional Ising model, for larger D
and/or larger ε the Monte Carlo data are strongly affected by problems of crossover between
different universality classes and, thus, ηr
p,crit(D) can be only estimated with rather modest
accuracy, allowing no firm statements about critical exponents. Approaching the transition
from ηr
p< ηr
p,crit(D), we find a strong increase of the correlation length ξ||describing the
correlation of interfacial fluctuations, but again the accuracy of our results would not suffice
to estimate the value of the associated critical exponent. In view of the fact that even for
the simple Ising model confined between competing boundaries a clarification of the critical
behavior turned out to be very difficult, the problems encountered for the present more
complicated model, which is strongly asymmetric even in the bulk, are not at all surprising.
The fact that observation of interface localization does not require very special conditions