Effect of Controlled Corrugation on Capillary Condensation of Colloid

Eﬀect of controlled corrugation on capillary condensation of colloid-polymer mixtures

Andrea Fortini∗ and Matthias Schmidt Theoretische Physik II, Physikalisches Institut, Universit¨atBayreuth, Universit¨atsstraße 30, D-95447 Bayreuth, Germany

We investigate with Monte Carlo computer simulations the capillary phase behaviour of model colloid-polymer mixtures confined between a flat wall and a corrugated wall. The corrugation is modelled via a sine wave as a function of one of the in-plane coordinates leading to a depletion attraction between colloids and the corrugated wall that is curvature dependent. We find that for increased amplitude of corrugation the region of the phase diagram where capillary condensation occurs becomes larger. We derive a Kelvin equation for this system and compare its predictions to the simulation results. We find good agreement between theory and simulation indicating that the primary reason for the stronger capillary condensation is an increased contact area between the fluid and the corrugated substrate. On the other hand, the colloid adsorption curves at colloid gas-liquid coexistence show that the increased area is not solely responsible for the stronger capillary condensation. Additionally, we analyse the dimensional crossover from a quasi-2D to a quasi-1D system and find that the transition is characterised by the appearance of a metastable phase.

I. INTRODUCTION tions the phase behaviour of a mixture of colloid and non- adsorbing polymers confined between a wrinkled wall and The equilibrium behaviour of a fluid in contact with a flat wall. We chose colloid-polymer mixtures because a solid substrate is ruled by the interfacial free energy, they have been extensively studied in the literature [17– i.e., the amount of free energy needed to create the in- 29], both from a theoretical and experimental point of terface [1]. Since this free energy is the product of the view. In particular, such mixtures allowed the investiga- interfacial tension and the total contact area, the fluid tion of a variety of fundamental inhomogeneous statisti- equilibrium behaviour at a substrate can be controlled cal phenomena such as wetting at single flat walls [30– by either manipulation of the chemical properties of the 34] and capillary condensation and evaporation between surface (interfacial tension) or the geometry of the sur- flat walls [35–40]. Here, we describe the colloid-polymer face (contact area). It is known that corrugated surfaces mixture using the Asakura-Osaawa-Vrij (AOV) [41, 42] have a higher contact area with respect to a flat surface model. Interestingly, the depletion attraction between with the same cross-sectional area [2, 3], and in nature, colloids and the corrugated wall depends on the local corrugation gives the Lotus flower [4] its characteristic curvature of the wall (the same effect occurs for mix- hydrophobicity. Efforts to mimic the Lotus effect resulted tures of hard spheres [43, 44]). We find that for in- in the production of micromachined surfaces whose wet- creased amplitude of corrugation the capillary conden- tability has been controlled by proper surface microstruc- sation is stronger, i.e. the region of the phase diagram turing [5, 6]. From a fundamental point of view wetting (phase space) where capillary condensation occurs be- and capillary condensation on structured [7] or curved comes larger. We also analyse the confined system us- substrates [8, 9] as well as on surfaces with wedge geome- ing a simple thermodynamic description, derive a Kelvin try [10, 11] have been studied in detail (see also the review equation and study its validity by comparing the theoret- by Bonn et al. [12] and references therein). Nevertheless, ical predictions to the simulation results. The agreement while a lot of research has been done for simple fluids, between theory and simulation is good, and indicate that the effect of corrugation has been neglected in complex the increased contact area between the fluid and the sub- colloidal fluids because the roughness of a substrate nor- strate (due to corrugation) is the primary cause for the mally occurs at length scales that are much smaller than stronger capillary condensation. A dramatic increase in the typical size of a colloidal particle. colloidal adsorption at the corrugated wall is also primar- Recently, a technique has been introduced that allows ily due to the increased surface area. On the other hand, arXiv:1301.7434v2 [cond-mat.soft] 22 Feb 2013 the controlled wrinkling of surfaces [13, 14] on the mi- further analysis suggests that other effects influence the cron scale. The effect of wrinkled substrates on crys- thermodynamics of the system, like for example the cur- tallisation has been investigated [15], and the resulting vature dependence of the particle wall interaction. Fur- structures have been used to enhance Surface Raman thermore, we find that the crossover from a quasi-2D to Spectroscopy [16]. Clearly, the wrinkling technique could a quasi-1D system leads to the emergence of metastable allow a systematic study of the effect of roughness on the phases characterised by a sequence of filled and empty phase and wetting behaviour of complex fluids. quasi-1D channels. In this article, we investigate with computer simula- The paper is organised as follow. In Sec. II we introduce the model and the simulation technique. In Sec. IV we derive an expression for the Kelvin equation for corrugated walls, while in Sec. V we present the simulation ∗ [email protected] results. Finally in Sec. VI we draw our conclusions. 2

� a) b)

H � � �

Vfree �p �c Ly �

Lx c) d)

Figure 1. a) Schematic drawing of the simulation box and model. Colloids of diameter σc and spheres with diameter σp representing the polymers are confined between hard walls: a flat wall at the bottom (z = 0) with area Aflat = Lx × Ly, and a sinusoidal wall of wavelength λ and amplitude a at the top (z = H + a sin(2πx/λ)) . b) Cross section of the simulation box. The depletion zones at the walls are delimited by dashed lines. The grey area indicates the volume accessible to the particles. c) Colloids and Polymer at the flat wall. The depletion zones are delimited by dashed lines. The grey area indicates the gain in free volume due to the overlap of two depletion regions. d) Like c) but for a colloid at a sinusoidal wall. The gain in free volume depends on the local curvature of the wall.

II. MODEL to the so-called depletion effect. In Fig. 1c), we show a colloid in contact with a wall. Around the colloid there is The Asakura-Osaawa-Vrij (AOV) [41, 42] model is a a depletion region (dashed circle) prohibited to the polymers due to the colloid-polymer hard-core interaction binary mixture of colloidal hard spheres of diameter σc (the chains cannot penetrate the colloids). The hard-core and of spheres with diameter σp representing polymer coils (Fig. 1a). The pair interaction between colloids is interaction between polymers and the wall gives rise to a depletion zone at the wall (dashed line in Fig. 1c). If two that of hard spheres: Vcc(r) = ∞ if r < σc and zero otherwise, where r is the separation distance between colloids approach each other, so that two depletion zones particle centres. The interaction between a colloid and overlap there is an increase in free volume for the polymer chains, i.e., an increase in entropy [17, 18, 41, 42]. a polymer is also that of hard spheres: Vcp(r) = ∞ if The increase in entropy can be described by an attractive r < (σc + σp)/2 and zero otherwise. The polymers, however, are assumed to be ideal, hence the polymer-polymer interaction between colloidal particles. Likewise, there is an increase in entropy when the depletion zones of col- interaction vanishes for all distances, Vpp(r) = 0. The loids and wall overlap (grey zone). The larger the ex- size ratio q = σp/σc = 1 is a geometric control parameter. cluded volumes, the stronger is the effective attraction The colloid-polymer mixture is confined between one between the hard wall and the colloids. Figure 1d) il- smooth planar hard wall at z = 0 and one sinusoidal hard lustrates the depletion zone of a curved wall. Clearly, wall at z = H + a sin(2πx/λ), where H is a measure of the gained free volume (grey zones) depends on the local the distance of the two walls, a is the amplitude of the curvature of the wall. The concave part of the wall leads sinusoidal corrugation, λ is its wavelength, and x is the to a larger gain in free volume (more attractive) than the lateral coordinate perpendicular to the wrinkles. Any convex part of the wall (less attractive). overlap between particles and walls is omitted, therefore We denote the packing fractions by ηi = the external potentials acting on species i = c, p 3 πσi Ni/(6AH), where Ni is the number of particles 0 r ∈ V of species i and A is the cross-sectional (normal to the V (z) = i free (1) ext,i ∞ otherwise, z-direction) area of the wrinkled wall, and also equal to the area of the flat wall. Additional thermodynamic where Vfree is the free volume available to the particles quantities are the scaled chemical potential βµc and βµp as sketched in Fig. 1b). of colloids and polymers, respectively, and the polymer r The polymers induce an effective colloid-colloid and reservoir packing fraction ηp = exp(βµp) of a reservoir of colloid-wall attraction that is of entropic origin and is due pure polymers that is in chemical equilibrium with the 3 system. The inverse temperature is β = 1/kBT , with kB The free energy in the grand canonical ensemble is the the Boltzmann constant and T the temperature grand potential Ω. For a fluid between a top and a bottom walls of area A1 and A2, respectively, the grand potential is the sum of the bulk free energy per unit vol- III. SIMULATION METHOD ume ω and the surface energy contributions from the two walls,

Fig. 1a) displays an illustration of the model, for which 1 2 Ω(µ) = V ω(µ) + A1γ (µ) + A2γ (µ) , (5) we carried out Monte Carlo computer simulations for a wide range of different values of particle concentrations where V is the volume, µ is the chemical potential and and for several values of the amplitude a. The wavelength γ1, γ2, are the fluid-wall interfacial tensions for the top of the corrugation was fixed to the value of λ = 10σc. Pe- and bottom wall, respectively. riodic boundary conditions were applied in the x- and y- We introduce the bulk chemical potential at coexis- directions, and the box size in the x-direction was chosen tence µb, and use it as a reference state for the chemical as 4λ. potential of the confined system We carried out Monte Carlo simulations in the grand canonical ensemble, i.e. with fixed volume, temperature, µ = µb + ∆µ. and chemical potentials βµc and βµp . To study the phase coexistence, we sample the proba- Assuming ∆µ small we Taylor expand equation (5) around µb, recalling that the bulk density is bility P (Nc)|µc,µp of observing Nc colloids in a volume V using the successive umbrella sampling [45]. We use the ∂ω histogram reweighing technique to obtain the probability ρ = − 0 ∂µ distribution for any µc once P (Nc)|µc,µp is known for a given µc: and that the adsorption is

0 ln P (Nc)|µ0 ,µ = ln P (Nc)|µ ,µ + Nc(βµ − βµc). (2) ∂γ c p c p c Γ = . ∂µ At phase coexistence, the distribution function P (Nc) becomes bimodal with two separate peaks of equal area The Taylor expansion yields for the colloidal liquid and gas phases. We determine Ω(µ) A1 1 which µ0 satisﬁes the equal area rule = ω(µb) − ρ(µb)∆µ + γ (µb) c V V A A A hN i ∞ 1 1 2 2 2 2 Z c Z + Γ (µb)∆µ + γ (µb) + Γ (µb)∆µ.(6) P (N )| 0 dN = P (N )| 0 dN , (3) V V V c µc,µp c c µc,µp c 0 hNci The coexistence between a liquid and a gas phase inside the slit occurs when both thermal and mechanical with the average number of colloids equilibrium conditions are satisﬁed, i.e., the chemical po- Z ∞ tential and pressure of the gas and liquid phases are the

hNci = NcP (Nc)|µc,µp dNc, (4) same. These two conditions are satisfied when the grand 0 potential of the two phases is the same Ωgas(µ) = Ωliq(µ). Given that ωliq(µ ) = ωgas(µ ), the above equilibrium using the histogram reweighing equation (2). The sam- b b condition inside the slit leads to the following relation pling of the probability ratio P (N)/P (N + 1) is done, in each window, until the difference between two successive 0 = (ρliq − ρgas)∆µ + samplings of the probability ratio is smaller than 1×10−4. A1 1 1 A1 1 1 To improve the sampling accuracy, we used the cluster + (γgas − γliq) + (Γgas − Γliq)∆µ + move introduced by Vink and Horbach [46]. V V A A + 2 (γ2 − γ2 ) + 2 (Γ2 − Γ2 )∆µ . (7) V gas liq V gas liq IV. THEORY: KELVIN EQUATION We stress that the above relation (7) is valid for any type of surface, as we only requested that ∆µ be small. The previous relation greatly simplifies if we further We apply a thermodynamic treatment in the limits of assume that the adsorption at the wall is negligible and λ/σ 1 and H/σ 1 to the system sketched in Fig. 1. that γ1 = γ2 = γ, in which case The derivation follows closely the derivation of Evans [47] for fluids between smooth parallel walls. For simplicity A1 A2 (γliq − γgas) we limit the theoretical derivation to the case of one- ∆µ = ( + ) (8) V V (ρliq − ρgas) component fluids. As will become clear at the end the section, the result can be generalised effortlessly to multi- We stress that the assumption γ1 = γ2 = γ implic- components fluids. itly considers a curvature independent interfacial tension. 4

a) where

20 A R = corr R Aﬂat

10 is the ratio between the flat and corrugated area and is equivalent to the Wenzel roughness ratio defined as the ratio between the contact area and the geometric cross 1 1 2 3 4 5 6 7 8 9 10 sectional area [2]. λ/σc By comparing Eqs. (9) and (10) we find also that the ratio between the chemical potential shifts of the corru- b) 4 gated and flat walls is related to the Wenzel ratio via 1 + R 3 ∆µcorr/∆µflat = . (11) R 2 2

1 C. Two corrugated walls 1 2 3 4 5 6 7 8 9 10 a/σc For the case of two corrugated walls the general Kelvin equation (8) becomes Figure 2. The roughness ratio R for a sinusoidal wall. a) For fixed amplitude a = 5σc and changing wavelength. b) For wavelength λ = 10σc and changing amplitude. 2 (γ − γ ) ∆µcorr = R liq gas , (12) hHi (ρliq − ρgas) That is, we treat the limit of a/σ 0. This appar- By comparing Eqs. (12) and (9) we find that for the case ently crude approximation is justified in the framework of two corrugated walls the ratio is of the Kelvin equation that treats systems in the limit

H/σ 1. Strictly speaking, the Kelvin equation is corr flat not valid for strongly confined systems, but has been ∆µ /∆µ = R. shown before that its predictions remain qualitatively good down to quasi-2D systems. This theoretical description would on the other hand break down in the quasi-1D D. Generalisation to binary mixtures limit, because it would predict a phase transition that in reality does not exists for a bulk one-dimensional system. The derivation of the Kelvin equation for binary mixtures [48] proceeds along the lines described above for a one-component fluid. A closed set of equation for A. Two flat walls the chemical potential shifts ∆µc, and ∆µp, for colloid and polymer, respectively, can however be obtained only If we consider two flat walls of equal area A1 = A2 = when an independent relationship between ∆µc, and ∆µp Aflat, and volume V = AflatH we obtain the standard is used. In Ref. 37 three possible choices for the rela- equation [47] tionship are outlined. Independent of the choice of the relationship, the equations for binary mixtures for corru- 2 (γ − γ ) ∆µflat = liq gas . (9) gated walls leads to H (ρliq − ρgas) 1 + R ∆µcorr/∆µflat = (13) c c 2 1 + R B. One corrugated wall and one flat wall ∆µcorr/∆µflat = , p p 2

A system with one flat wall and one corrugated wall and has a volume V = AflathHi, with hHi the average separation distance between the walls. The relation (8) then corr flat ∆µc /∆µc = R (14) becomes corr flat ∆µp /∆µp = R, 1 (γ − γ ) ∆µcorr = (1 + R) liq gas , (10) hHi (ρliq − ρgas) for one and two corrugated walls, respectively. 5

E. Roughness ratio for a sinusoidal wall 1.5 We consider a wall corrugated in one direction by a sinusoidal functional form as shown in the sketch of 1.4 gas Fig. 1a). The cross sectional area is Aﬂat = Lx × Ly, where Lx and Ly are the parallel and perpendicular directions with respect to the sinusoidal direction, and 1.3 Lx = nλ, where n is an arbitrary number and λ is the

wavelength. Therefore, the roughness ratio can be writ- r p ten as η 1.2 bulk 3D q σ R nλ 4π2 2 2π Flat H=5 c 0 dx 1 + λ2 cos ( λ x) R = , (15) 1.1 a=0 σc nλ a=5 σc

a=7 σc where the numerator is the line integral over the sinu- 1.0 liquid a=9 σc soidal path. The elliptical integral can be solved numer- a=11 σc ically. The value of the integral (15) is the same for any a=15 σc integer value of n, but changes when non-integer values 0.9 of n are chosen. Therefore, an explicit dependence of the 3.6 3.8 4.0 4.2 4.4 4.6 4.8 integral on n is left for the general case. βµc Figure 2 shows the behaviour of the roughness ratio defined in Eq. (15). For fixed amplitude and increasing wave length (Fig. 2a) the ratio R decreases. Clearly for Figure 3. Phase diagram in the (βµp, βµc) representation. λ → ∞ the ratio goes to one. On the other hand, for fixed Shown are the binodals for wavelength λ = 10σc and amplitudes a/σc=0, 5, 7, 9, 11, 13, and 15. Also shown are the wavelength λ (Fig. 2b) the roughness ratio R increases binodals for the bulk system (dashed line) and for the system monotonically for increasing values of the amplitude of confined between two flat walls at distance H = 5σc (thick corrugation a. continuos line).

V. SIMULATION RESULTS sinusoidal wall, the binodals shifts toward larger polymer chemical potentials and smaller colloid chemical poten- A. Phase diagram tials, indicating progressively stronger capillary condensation with respect to the flat walls, i.e., larger region Figure 3 shows the phase diagram in the (βµp, βµc) of phase space where the capillary condensation occurs. representation. Each line represent a binodal, i.e. the The critical points were not calculated because an ac- value of colloid and polymer chemical potentials at which curate determination would require a careful finite-size phase separations occurs. For chemical potentials in the analysis [38] that is beyond the scope of the current work. region above a binodal the gas phase is stable, whereas The end-points in Fig. 3 were determined as the lowest in the region below the binodal the liquid phase is sta- value of the chemical potentials at which a double peak ble. In particular, the capillary binodals for corrugation in the probability distribution was observed. amplitude a/σc = 0 − 15 are shown (symbols). As refer- One trivial interpretation for the behaviour is that the ence we also show the bulk binodal (dashed line), and the sinusoidal wall introduces a length scale in the system capillary binodal for flat walls with separation distance that is smaller than the average wall separation distance, H/σc = 5 (thick continuous line). The binodal for the namely hmin = H −a, that strengthens the capillary con- colloid-polymer mixtures confined between two flat walls densation. However, this interpretation is not supported (i.e., a = 0 H/σc = 15) is shifted toward smaller col- by our results. For example for a = 11σc, the smallest loidal chemical potentials and higher polymer chemical length scale is hmin = 15σc − 11σc = 4σc. As a refer- potential with respect to the bulk binodal (thin dashed ence in Fig. 3 we plot the binodal for two flat walls at line). The shift of the binodals indicate the occurrence distance H/σc = 5. The a = 11σc binodal (left triangles) of capillary condensation, because in the region between is clearly separated from it. the confined and bulk binodals, the stable phase for the For amplitude a = 15σc the system consist of a se- confined system is the liquid, while for the bulk system is ries of independent quasi-1D channels. Interestingly, we the gas. This effect was extensively studied in the litera- find that the binodal lines for increasing amplitudes a, ture [35, 37, 38, 49] for colloid-polymer mixtures confined slowly approach the binodal of the system with a single between flat walls. groove, therefore the system undergoes a slow dimen- Here, for colloid-polymer mixtures confined between sional crossover from a quasi-2D to a quasi-1D system. one flat and one corrugated wall, we find that for in- We stress that in one-dimensional systems the gas-liquid creasing corrugation, i.e., increasing amplitude a of the phase transition does not exist in the thermodynamic 6

C. Adsorption 3.0 r ηp =1.0 r Further insights can be gained by calculating the col- ηp =1.1 loid adsorption Γ for state points at gas coexistence. First r 2.5 ηp =1.2 of all, we carried out simulations of colloid-polymer mix- r tures between two ﬂat walls at separation H = 15σc to t ηp =1.3 a

l measure the adsorption at one ﬂat wall f η r =1.4

µ p 2.0 N flat − N bulk ∆ Γ = c c (16) / flat flat

r 2A r o c at the gas coexistence state points of the phase diagram µ 1.5 shown in Fig. 3. Subsequently, we performed simulations ∆ for the mixtures confined between one flat and one sinusoidal wall and computed the adsorption at a single sinusoidal wall 1.0 corr bulk Nc − Nc Γcorr = flat − Γflat . (17) 0 2 4 6 8 10 12 A a/σc Here we choose as reference the cross-sectional area Aflat, as it is common for corrugated surfaces where the real contact area is not easily known. We find that the adsorp- corr flat Figure 4. The ratio ∆µc /∆µc for different amplitudes tion increases dramatically with increasing amplitudes a a of the sinusoidal wall at fixed wavelength λ = 10σc. We and changes little for increasing polymer reservoir pack- compare results of the computer simulations (symbols) for r ing fractions (Fig. 5a). different values of polymer reservoir packing fraction ηp with In order to evaluate the effect of the chosen reference the theoretical prediction (line) of equation (10). area, we also calculated the adsorption using as a reference the real contact area Acorr, i. e. limit. Even in quasi-1D pores a rounding of the tran- N corr − N bulk Γ∗ = c c − Γ . (18) sition is noted [50] and a multi domain structure is ob- corr Acorr flat served instead of a proper gas-liquid separation. The increase in colloid adsorption for increasing amplitudes a is now less dramatic (Fig. 5b), note the different y-axes scale) but still clearly visible. Therefore, the in- B. Comparison with the Kelvin equation creased surface area of the sinusoidal wall is not the only factor responsible for the increased adsorption and con- The Kelvin equation (11) suggests, on the other hand sequently for the stronger capillary condensation. that the stronger capillary condensation for increasing Interestingly, the simulation snapshots for state points amplitudes a is due to the increased surface area of the at gas coexistence for polymer reservoir packing fraction r corrugated wall. To evaluate the validity of this inter- ηp = 1.4 (shown in Fig. 6) show that colloids are adsorbed pretation we compare theoretical results with the sim- inside the grooves of the sinusoidal walls, that is in the ulation findings. In particular, we compare the ratio concave parts of the corrugated wall. corr flat ∆µc /∆µc found in simulations to the theoretical result of Eq. (11). Remarkably, the simulation results start at smaller values of the ratio but they follow the D. Free energy curves same trend as the theory results for increasing amplitudes a/σc, demonstrating that the stronger capillary conden- We next explore the dimensional crossover that oc- sation is mainly driven by the increased surface area of curs at amplitudes a/σc comparable to the wall dis- the corrugated wall. We believe that the smaller values tance H. Consider that the free energy (grand poten- corr flat of the ratio ∆µc /∆µc in simulations with respect to tial) difference with respect to a reference state is given the theory are due to the large adsorption of colloids at by − ln(P (ηc)|µc,µp ), where P (ηc)|µc,µp ) is the probabil- the sinusoidal wall in the gas phase. The Kelvin equa- ity of observing the system at colloid packing fraction ηc. tion also neglects the curvature. Recent works on curved In Fig. 7a) we plot the free energy curves for amplitudes surfaces [51, 52] suggest that the inclusion of curvature a=1, 3, 5, 7 σc. The two minima correspond to the gas effects would lead to smaller values of the γliq − γgas dif- and liquid coexisting densities. For intermediate pack- ference. This leads to a decrease of the shift in chemical ing fractions ηc the free energy decreases for increasing potential predicted by the Kelvin equation, leading to a packing fraction, as expected for systems in slit geom- better agreement with simulation results. etry. In Fig. 7b) the free energy curves for amplitudes 7

a) b) a) 4.5

4.0

3.5

3.0 c) d) a/σc =1 r r

o a/σ =3 c 2.5 c Γ a/σc =5

2.0 a/σc =7 a/σ =9 1.5 c a/σc =11 1.0 a/σc =13 a/σc =15 Figure 6. Snapshots of the simulation at gas-liquid coex- 0.5 istence for the packing fraction corresponding to the equi- 1.0 1.2 1.4 1.6 1.8 librium gas phase and polymer reservoir packing fraction r r ηp ηp = 1.4. The colloidal particles (cyan/light grey spheres) and the polymers (green/dark grey spheres) are confined be- b) 1.4 tween one flat wall (bottom) and one sinusoidal wall (top). The walls occupy the white sections of the figures. a) Ampli- 1.3 tude a/σc = 1 b) Amplitude a/σc = 5 c) Amplitude a/σc = 9 d) Amplitude a/σ = 13 1.2 c 1.1

a/σc =1 tures [53–55]. r

r 1.0 o

∗ a/σ =3 c c Γ 0.9 a/σc =5 a/σ =7 VI. SUMMARY AND CONCLUSIONS 0.8 c a/σc =9 0.7 a/σc =11 We traced the phase diagram of model colloid-polymer a/σ =13 0.6 c mixtures confined between one flat wall and one corru- a/σc =15 gated wall with grand-canonical MC simulations. The 0.5 corrugation was modelled by a sinusoidal function of am- 1.0 1.2 1.4 1.6 1.8 plitude a. We found that for increasing values of a the r ηp capillary condensation get stronger, i.e., the region of parameter space where the capillary condensation oc- Figure 5. Adsorption at the corrugated wall for increasing curs becomes larger. We derive a Kelvin equation for values of the corrugation amplitude a/σc. a) Colloidal ad- the system that predicts that capillary condensation in flat Acorr sorption Γcorr with the cross-sectional area A as reference the system is controlled by the Wenzel ratio R = A ∗ flat surface. (Eq. (17)). b) Colloidal adsorption Γcorr with the between areas of the corrugated and flat walls. We find real contact area Acorr as reference surface (Eq. (18)). very good agreement between simulation results and the Kelvin equation prediction, indicating that the increased contact area between the fluid and the substrate (due to corrugation) is the primary cause for the stronger capil- a=9, 11, 13, 15 σc are shown. The approaching dimen- lary condensation. The analysis of the adsorption of col- sional crossover is characterised by the appearance of a loidal particles shows a strong preference of the colloids sequence of minima in the free energy curves. In our to adsorb at the corrugated wall in agreement with the system the local minima appear first for a/σc = 9 and stronger capillary condensation. The finding is also cor- becomes stronger and stronger for increasing amplitudes. roborated by a visual inspection of the simulation snap- These minima indicate the presence of metastable phases shots, which show a strong adsorption of colloids in the characterised by coexistence of channels filled with fluid grooves of the corrugated walls. Nevertheless, we find phase and channels containing the gas phase. Each time an increase of adsorption for increased surface area also a channel is completely filled we find an energy minimum when the real contact area is used as reference. One pos- due to the decreased interfacial energy as shown in Fig. 8 sible explanation for this effect is that the depletion effect r for polymer reservoir packing fraction ηp = 1.4 and cor- responsible for the colloid-wall effective attractive inter- rugation amplitude a/σc = 11. These phases are similar action is curvature dependent. The overlap area between to the zebra phase predicted for optically confined mix- the particle and wall excluded volumes is different for 8

a) 200 teraction. We also observe a dimensional crossover from a quasi-2D system to a quasi-1D system. The quasi-1D a =σ c system occurs when the amplitude of the corrugated wall is equal to the average separation distance, i.e., all chan- 150 a =3σc nels are decoupled. The free energy curves show that the approaching crossover is characterised by the appearance of a metastable phase, with partially ﬁlled channels, sim- )

P a =5σ ilar to the ’zebra’ phase predicted for optically conﬁned ( 100 c n colloid-polymer mixtures [53, 54]. l

− In this paper we only considered capillary condensation a =7σc a) b) 50

0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ηc c) d)

b) 100

60 ) a =9σ P c

( Figure 8. Snapshots of the simulation at polymer reservoir n

l r 40 a =11σc packing fraction ηp = 1.4 and amplitude a/σc = 11. The − α colloidal particles (cyan/light grey spheres) and the polymers β (green/dark grey spheres) are conﬁned between one ﬂat wall 20 γ (bottom) and one sinusoidal wall (top). The walls occupy

a =13σc the white sections of the figures. a) One channel filled with colloids at state point α in Fig. 7b). b) Two channels filled 0 a =15σc with colloids at state point β in Fig. 7b). c) Three channels 0.00 0.05 0.10 0.15 0.20 0.25 0.30 filled with colloids at state point γ in Fig. 7b). d) Equilibrium ηc liquid phase.

Figure 7. Logarithm of the probability distribution (free en- r and neglected two other relevant phenomena, namely ergy) at polymer reservoir packing fraction ηp = 1.4 and colloid chemical potentials at coexistence. a) For amplitudes wetting [12] and wedge filling transitions [11, 56, 57]. a=1, 3, 5, 7 σc. b) For amplitudes a=9, 11, 13, 15 σc. Given the finite system size as well as the finite wedge geometry our assumption is completely reasonable, but an extension of this work would be to analyse the wet- different wall curvatures. Therefore the wall is more at- ting and wedge filling behaviour of a single corrugated tractive at the concave side and less attractive at the con- wall and compare to works on wedge filling transition in vex edge. In our simulation we used a fixed wavelength finite geometries [58–60]. and changing amplitudes, i.e., the curvature of the corru- A recent paper [55] on fluids in periodic confinement gated walls would be different for the different amplitudes showed unexpected correlations between the channels leading to a different effective wall-colloid attractive in- that deserve further investigation in the current system.

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