Critical Casimir effect for colloids close to chemically patterned substrates
aa r X i v : . [ c ond - m a t . s o f t ] M a y Critical Casimir effect for colloids close to chemically patterned substrates
M. Tröndle,
1, 2
S. Kondrat,
1, 2, a) A. Gambassi, L. Harnau,
1, 2 and S. Dietrich
1, 2 Max-Planck-Institut für Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart,Germany SISSA — International School for Advanced Studies and INFN, via Bonomea 265, 34136 Trieste,Italy (Dated: May, 6 2010)
Colloids immersed in a critical or near-critical binary liquid mixture and close to a chemically patterned substrate aresubject to normal and lateral critical Casimir forces of dominating strength. For a single colloid we calculate theseattractive or repulsive forces and the corresponding critical Casimir potentials within mean-field theory. Within thisapproach we also discuss the quality of the Derjaguin approximation and apply it to Monte Carlo simulation dataavailable for the system under study. We find that the range of validity of the Derjaguin approximation is rather largeand that it fails only for surface structures which are very small compared to the geometric mean of the size of thecolloid and its distance from the substrate. For certain chemical structures of the substrate the critical Casimir forceacting on the colloid can change sign as a function of the distance between the particle and the substrate; this provides amechanism for stable levitation at a certain distance which can be strongly tuned by temperature, i.e., with a sensitivityof more than 200nm / K.PACS numbers: 05.70.Jk, 82.70.Dd, 68.35.Rh
I. INTRODUCTION
Since the discovery of the Casimir effect in quantumelectrodynamics it is well-known that the inherent fluctua-tions of a medium lead to an effective force acting on its con-fining boundaries. In soft matter physics, the analogue of thevacuum fluctuations in quantum electrodynamics are the ther-mal fluctuations of the order parameter f of a fluid. These oc-cur on the length scale of the bulk correlation length x whichis generically of molecular size. However, upon approach-ing a critical point at the temperature T = T c , the correlationlength x increases with an algebraic singularity and attains macroscopic values. The confinement of these long-rangedfluctuations results in the so-called critical Casimir force act-ing on a length scale set by x . Since the correlation length di-verges as x ( T → T c ) (cid:181) | T − T c | − n , where n is a standard bulkcritical exponent, the range of the critical Casimir force (andtherefore its strength at a certain distance) can be controlledand tuned by minute temperature changes (see, e.g., Refs. 4and 5). The characteristic energy scale of the critical Casimireffect is given by k B T c , which allows for a direct measurementof the critical Casimir forces, in particular if the critical pointis located at ambient thermodynamic conditions .The attractive or repulsive character of the critical Casimirforce can be controlled by suitable treatments of the confiningsurfaces. Generically, the surfaces which confine a binary liq-uid mixture preferentially adsorb one of its two components(or the gas or liquid phase in the case of a one-componentfluid). This can be described by effective, symmetry breakingsurface fields, which lead to a preference for either positive [(+)] or negative [( − )] values of the scalar order parameter f , corresponding to the difference between the local concen-trations of the two species (or the deviation of the density ofthe one-component fluid from its critical value). The criti-cal Casimir force strongly depends on the effective boundary a) Present address: Department of Chemistry, Imperial College London, SouthKensington Campus, SW7 2AZ London, U.K. conditions (BC) at the walls (see, e.g., Refs. 8–15 and refer-ences therein). It is attractive for equal symmetry breaking ( ± , ± ) BC and repulsive for opposing ( ± , ∓ ) BC. Inter alia,this latter feature qualifies critical Casimir forces to be a toolto overcome the problem of “stiction” which occurs in micro-and nano-mechanical devices. (The quantum electrodynamicCasimir force is typically attractive and thus responsible forstiction; turning it to be repulsive requires a careful choice ofthe fluid and of the bulk materials of the confinement .) Thetheoretical description of the critical Casimir forces is partic-ularly challenging due to the non-Gaussian character of theorder parameter fluctuations, which contrasts with the intrin-sically Gaussian nature of the low energy fluctuations of theelectromagnetic field; in addition, the critical Casimir effect isalso particularly rich as it allows, inter alia, symmetry break-ing boundary conditions, which do not occur for electromag-netic fields.The critical Casimir effect exhibits universality, i.e., thecritical Casimir force expressed in terms of suitable scalingvariables depends only on the universality class of the bulkcritical point and on the type of boundary conditions, whereasit is independent of the microscopic structure and of the ma-terial properties of the specific fluid medium involved. In ourpresent theoretical analysis we focus on the Ising universalityclass which encompasses the experimentally relevant classicalbinary liquid mixtures and simple fluids.The existence of the critical Casimir effect has been exper-imentally confirmed and its strength has been first measuredindirectly for wetting films . The first direct measurementof this effect has been performed at the sub-micrometer scalefor a spherical colloid immersed in a (near) critical binaryliquid mixture close to a laterally homogeneous and planarsubstrate . The corresponding Monte Carlo simulation datafor the film geometry are in very good quantitative agreementwith all available experimental data . Theoretical stud-ies of the critical Casimir effect acting on colloidal particlesinvolve spherically or ellipsoidally shaped colloids ad-jacent to homogeneous substrates.Besides their wide use as model systems in soft matter Y = P | Q | = P / p R x ± l = L / P Q = sign ( t ) D / x ± L = L / √ RD P = P / √ RD homogeneouschemical step (s)single chemical lane ( ℓ )periodic pattern (p) D = D / R X = X / √ RD colloid X = X = X = ( a > )( a ) x = X substrate DR ( b ) x P ( a < ) ( a ℓ )( a ) ( a )( a )( a ) L L LL FIG. 1. Sketch of a spherical colloid immersed in a near-critical bi-nary liquid mixture (not shown) and close to a (patterned) planar sub-strate. The sphere with ( b ) boundary condition (BC) and radius R islocated at a surface-to-surface distance D from the substrate and itscenter has a lateral coordinate x = X with the substrate pattern beingtranslationally invariant in all other directions. The following fourdifferent types of substrate surfaces are considered: homogeneoussubstrate [Sec. III], a chemical step [s; Sec. IV], a single chemicallane [ ℓ ; Sec. V], and a periodically patterned substrate [p; Sec. VI].(Note that for a four-dimensional system, which we also consider,this is a three-dimensional cut of the system, which is invariant alongthe fourth direction; the sphere thus corresponds to a hypercylinderin four dimensions.) For later reference, the box on the left sidesummarizes the definitions of the various scaling variables which thescaling functions of the critical Casimir force depend on for the listedgeometrical configurations. On the right, ( a ) , ( a ≷ ) , ( a ℓ ) , ( a ) , and ( a ) indicate the boundary conditions corresponding to the variouschemical patterns. physics, colloids have applications at the micro- and nanome-ter scale. In this context, they are widely used in micro- andnano-mechanical devices. Therefore, one may utilize the crit-ical Casimir forces acting on colloids because their strengthand their direction can be tuned in a controlled way. Suitablydesigned chemically or geometrically structured substratesgenerate lateral critical Casimir forces acting on colloidalparticles . Current techniques allow one to endow solidsurfaces with precise structures on the nano- and micrometer-scale. Hence, the critical Casimir effect can be used to createlaterally confining potentials for a single colloid, which canbe tuned by temperature .Recently, the critical Casimir potential of a colloid close toa substrate with a pattern of parallel chemical stripes with lat-erally alternating adsorption preference has been measured .In our corresponding theoretical study , we have calculatedthe normal and lateral critical Casimir forces acting on a col-loid close to such a patterned substrate as well as the corre-sponding potentials. We have used our theoretical predictionsfor the universal scaling functions of the critical Casimir po-tential in order to interpret the available experimental data inRef. 32. It has turned out that an agreement between theoryand experiment can be achieved only if one takes into accountthe geometrical details of the chemical substrate pattern. This demonstrates that the critical Casimir effect is very sensitiveto the details of the imprinted structures and that it can resolvethem.Here we generalize our previous analysis to various sub-strate patterns. In particular we study the critical Casimir ef-fect for a three -dimensional sphere close to a homogeneoussubstrate [Sec. III], a chemical step [Sec. IV], a single chemi-cal lane [Sec. V], and periodic patterns of chemical stripes ofalternating adsorption preference [Sec. VI] [see Fig. 1]. Forcompleteness, we also consider a cylinder which is alignedwith the chemical pattern [Sec. VII]. We provide quantitativepredictions for the scaling functions of the critical Casimirforces, pursuing a two-pronged approach: (i) We calculatethe force using the full three-dimensional numerical analysisof the appropriate mean-field theory (MFT). (ii) We use theso-called Derjaguin approximation (DA) based on the scal-ing functions for the critical Casimir force in the film geom-etry either obtained analytically within MFT or obtainedfrom Monte Carlo simulations , which allows us to pre-dict the critical Casimir force in the physically relevant three-dimensional case. Inter alia, we determine the range of valid-ity of the DA within MFT, which provides guidance concern-ing its applicability in three spatial dimensions d =
3. This isan important information because presently available MonteCarlo simulations are far from being able to capture complexgeometries .Currently, the possibility of realizing stable levitation ofparticles by means of the electrodynamic Casimir forces hasbeen the subject of intense theoretical investigation . Ourresults presented in Secs. VI and VII show that for suitablechoices of the geometry of the chemical pattern of the sub-strate, the critical Casimir forces can be used to levitate a col-loid above the substrate at a height which can be tuned bytemperature. This levitation is stable against perturbations be-cause it corresponds to a minimum of the potential of the crit-ical Casimir force acting on the colloid.In Sec. II we briefly introduce the necessary terminologyrelated to finite-size scaling and we discuss briefly the corre-sponding MFT. Section III is devoted to the well-studied caseof a colloid close to a homogeneous substrate. (In d =
4, as ap-propriate for MFT, the three-dimensional colloid is extendedto the fourth dimension as a hypercylinder, for which we alsopresent the results of our analysis.) As mentioned above, thevarious patterns and setups are considered in Secs. IV–VII.We conclude and summarize our findings in Sec. VIII. Cer-tain important technical details concerning the calculation ofthe Derjaguin approximation are presented in the AppendicesA–D.
II. THEORETICAL BACKGROUNDA. Finite-size scaling
According to the theory of finite-size scaling, the normaland lateral critical Casimir forces and the corresponding po-tentials can be described by universal scaling functions, whichare independent of the molecular details of the system butdepend only on the gross features of the system, i.e., on thebulk universality class (see, e.g., Refs. 8 and 9 and referencestherein) of the associated critical point. Here, we focus on theIsing universality class (which is characterized by a scalar or-der parameter f ) in spatial dimensions d = d =
4. In ad-dition, the critical Casimir force depends on the type of effec-tive boundary conditions at the walls, which we denote by ( a ) and ( b ) , and by the geometry of the confining surfaces .Note that ( a ) and ( b ) can represent the various symmetry pre-serving fixed-point BC (the so-called ordinary, special, peri-odic, or antiperiodic boundary conditions ) in addition to thesymmetry breaking cases ( ± ) we are mainly interested in, andwhich describe the adsorption of fluids at the confining walls.Inspired by the experiments described in Ref. 32 we con-sider binary liquid mixtures with their consolute critical pointapproached by varying the temperature T towards T c at fixedpressure and critical composition. We first study the film ge-ometry in which the fluid undergoing the continuous phasetransition is confined between two parallel, infinitely extendedwalls at distance L . According to renormalization group the-ory the normal critical Casimir force f ( a , b ) per unit area whichis acting on the walls scales as f ( a , b ) ( L , T ) = k B T L d k ( a , b ) ( sign ( t ) L / x ± ) , (1)where ( a , b ) denotes the pair of boundary conditions ( a ) and ( b ) characterizing the two walls. The scaling function k ( a , b ) depends only on a single scaling variable given by the signof the reduced temperature distance t from the critical point( ± for t ≷
0) and the film thickness L in units of the bulkcorrelation length x ± ( t → ± ) = x ± | t | − n , where n ≃ .
63 in d = n = / d = . (Clearly, one has f ( a , b ) ( L , T ) = f ( b , a ) ( L , T ) .) Positive values of t , t >
0, correspond to thedisordered (homogeneous) phase of the fluid, whereas neg-ative values of t , t <
0, correspond to the ordered (inhomo-geneous) phase, where phase separation occurs. Typically,the homogeneous phase is found at high temperatures, andone has t = ( T − T c ) / T c . However, many experimentally rel-evant binary liquid mixtures exhibit a lower critical point,for which the homogeneous phase corresponds to the low-temperature phase and one has t = − ( T − T c ) / T c . Thetwo non-universal amplitudes x ± of the correlation lengthare of molecular size and characterized by the universal ratio x + / x − ≃ . d = and x + / x − = √ d = ; x ± isdetermined by the exponential spatial decay of the two-pointcorrelation function of the order parameter f in the bulk.At the critical point T = T c , the correlation length diverges, x ± → ¥ , and the scaling function of the critical Casimir forceacting on the two planar walls attains a universal constantvalue referred to as the critical Casimir amplitude : k ( a , b ) ( L / x ± = ) = D ( a , b ) . (2)Away from criticality, the critical Casimir force decays ex-ponentially as a function of L / x ± . For the specific case ofsymmetry breaking BC a , b ∈ { + , −} and for t > L / x + ≫ pure exponential decay of f (+ , ± ) (see,e.g., Refs. 11, 35, and 48 and footnote 3 in Ref. 31, i.e., adecay without an algebraic prefactor to the exponential andwithout a numerical prefactor to L / x + in the argument of theexponential) corresponding to k (+ , ± ) ( L / x + ≫ ) = A ± (cid:18) L x + (cid:19) d exp ( − L / x + ) , (3) where A ± are universal constants . Note that, in the absenceof symmetry-breaking fields inside the film, the scaling func-tions for (+ , +) BC are the same as for ( − , − ) BC.
B. Mean-field theory
The standard Landau-Ginzburg-Wilson fixed-point effec-tive Hamiltonian describing critical phenomena of the Isinguniversality class is given by H [ f ] = Z V d d r (cid:26) ( (cid:209) f ) + t f + u f (cid:27) , (4)where f ( r ) is the order parameter describing the fluid, whichcompletely fills the volume V in d -dimensional space. Thefirst term in the integral in Eq. (4) penalizes local fluctuationsof the order parameter. The parameter t in Eq. (4) is propor-tional to t , and the coupling constant u is positive and providesstability of the Hamiltonian for t <
0. The mean-field orderparameter profile m : = u / h f i minimizes the Hamiltonian,i.e., δ H [ f ] / δ f | f = u − / m =
0. In the bulk the mean-field or-der parameter is spatially constant and attains the values h f i = ± a | t | b for t < h f i = t >
0, where, besides x + , a isthe only additional independent non-universal amplitude ap-pearing in the description of bulk critical phenomena , and b ( d = ) = / t = t ( x + ) − and u = a ( x + ) − . In a finite-size system thebulk Hamiltonian H [ f ] is supplemented by appropriate sur-face and curvature (edge) contributions . In the strong ad-sorption limit , these contributions generate boundary con-ditions for the order parameter such that f (cid:12)(cid:12) surface = ± ¥ . Forbinary liquid mixtures these fixed-point ( ± ) BC are the exper-imentally relevant ones. (Note that a weak adsorption prefer-ence might lead to a crossover between various kinds of effec-tive boundary conditions for the order parameter f .)We have minimized numerically H [ f ] using a 3 d finiteelement method in order to obtain the (spatially inhomoge-neous) profile m ( r ) for the geometries under consideration[see Fig. 1]. The normal and the lateral critical Casimir forcesare calculated directly from these mean-field order parameterprofiles using the stress tensor . This allows one to inferthe universal scaling functions of the critical Casimir forces atthe upper critical dimension d = (cid:181) u − and up to logarithmic corrections. The correspondingcritical Casimir potential is obtained by the appropriate inte-gration of the normal or of the lateral critical Casimir forces.In the case of planar walls the MFT scaling functions forthe critical Casimir force can be determined analytically and one finds [see Eq. (2)] for the case of symmetry break-ing boundary conditions the following critical Casimir ampli-tudes: D (+ , +) = D ( − , − ) = [ K ( / √ )] / u ≃ − . × u − ,where K is the complete elliptic integral of the first kind, and D (+ , − ) = − D (+ , +) [see Ref. 35 and Eq. (27) and Ref. [49] inRef. 23].In d = III. HOMOGENEOUS SUBSTRATE
We first consider a three-dimensional sphere of radius R with ( b ) BC facing a chemically homogeneous substrate with ( a ) BC at a surface-to-surface distance D as shown in Fig. 1,denoting this combination by ( a , b ) . The critical Casimirforce F ( a , b ) ( D , R , T ) normal to the substrate surface and thecorresponding critical Casimir potential F ( a , b ) ( D , R , T ) = R ¥ D d z F ( a , b ) ( z , R , T ) take the scaling forms F ( a , b ) ( D , R , T ) = k B T RD d − K ( a , b ) ( Q , D ) (5)and F ( a , b ) ( D , R , T ) = k B T RD d − J ( a , b ) ( Q , D ) , (6)where D = D / R and Q = sign ( t ) D / x ± (for t ≷
0) are the scal-ing variables corresponding to the distance D in units of theradius R of the colloid and of the correlation length x ± , respec-tively. The case d = D d − , which for d = F ( a , b ) and F ( a , b ) per length L of the extra translationally invariantdirection of the hypercylinder. A. Derjaguin approximation
The Derjaguin approximation (DA) is based on the idea ofdecomposing the surface of the spherical colloid into infinitelythin circular rings of radius r and area d S ( r ) = pr d r whichare parallel to the opposing substrate surface . (Herewe do not multiply 2 pr d r by the linear extension L of thehypercylinder along its axis in the fourth dimension, becausethe critical Casimir force is eventually expressed in units of L , which therefore drops out from the final expressions.) Thedistance L of a ring with radius r from the substrate is givenby L ( r ) = D + R (cid:18) − q − r / R (cid:19) . (7)Assuming additivity of the forces and neglecting edge effects,the normal critical Casimir forces d F ( r ) acting on these ringscan be expressed in terms of the force acting on parallel plates[Eq. (1)]: d F ( r ) k B T = d S [ L ( r )] d k ( a , b ) ( sign ( t ) L ( r ) / x ± ) . (8)Finally, in order to calculate the total force F ( a , b ) acting onthe colloid, one sums up the contributions of the rings, whichyields F ( a , b ) ( D , R , T ) k B T ≃ p Z R d rr [ L ( r )] − d k ( a , b ) ( sign ( t ) L ( r ) / x ± ) . (9)(For d = F ( a , b ) is the force on a sphere whereas in d = Q = sign ( t ) D / x ± (a) (+ , − ) repulsive ( − , − ) attractive K ( ∓ , − ) ( Q , D ) . (cid:12)(cid:12) K ( − , − ) ( , ) | DA ( d = d = D = / D = D J ( Q , D ) . | J ( − , − ) ( , ) | (b) Q = sign ( t ) D / x ± DA ( d = d = D = / D = FIG. 2. (a) Scaling functions K ( ∓ , − ) for the normal critical Casimirforce [Eq. (5)] acting on a three-dimensional sphere with ( b ) = ( − ) BC close to a homogeneous substrate with ( a ) = ( ∓ ) BC [Fig. 1].The suitably normalized scaling functions K ( ∓ , − ) are shown as afunction of the scaling variable Q = sign ( t ) D / x ± for t ≷
0, where t isthe reduced deviation from the critical temperature and K ( − , − ) ( , ) is the value of the critical Casimir force scaling function within theDA at T = T c for ( − , − ) BC. The solid lines correspond to the Der-jaguin approximation (DA, D = D / R →
0) within mean-field theory(MFT, d =
4) whereas the dotted lines correspond to the DA obtainedby using Monte Carlo (MC) results for films in d = . The normalization im-plies that at Q = − ( − , − ) BC whereas the solid line passes through 4 for (+ , − ) BC.The symbols correspond to the full numerical MFT results obtainedfor D = / D =
1, the size of which indicates the estimatednumerical error. (For (+ , − ) BC and t < K ( ∓ , − ) on D drops out, the difference between the symbols ⊡ and ⊙ and thesolid lines measures the accuracy of the DA in d =
4. (b) Differ-ence
D J = J (+ , − ) − J ( − , − ) of the scaling functions for the Casimirpotentials [Eq. (6)] for (+ , − ) and ( − , − ) BC, suitably normalizedby J ( − , − ) ( , ) . The solid line corresponds to the DA within MFTand the symbols correspond to the full MFT results for D = / D =
1; the dotted line is the DA for d =
3. Due to the normalizationthe solid line reaches 5 for Q = One expects the DA to describe the actual behavior accu-rately if the colloid is very close to the substrate, i.e., for D = D / R →
0. In this limit, Eq. (7) can be approximated by L ( r ) = D a where a = + r / ( RD ) , so that one finds forthe scaling function of the force K ( a , b ) ( Q , D → ) = p Z ¥ d aa − d k ( a , b ) ( a Q ) , (10)and, accordingly, for the scaling function of the potential J ( a , b ) ( Q , D → ) = p Z ¥ d b (cid:18) b d − − b d (cid:19) k ( a , b ) ( b Q ) . (11)At the bulk critical point, using Eq. (2), one findsthe well known values K ( a , b ) ( , ) = p D ( a , b ) / ( d − ) and J ( a , b ) ( , ) = p D ( a , b ) / [( d − )( d − )] . We note that the DAimplies that the dependence of F ( a , b ) and F ( a , b ) on the size R of the sphere reduces to the proportionality (cid:181) R indicatedexplicitly in Eqs. (5) and (6).Before proceeding further one first has to assess the accu-racy of the DA, which will carried out below within MFT( d = d =
3, so that within that range one can use theDA based on scaling functions for the film geometry obtainedfrom Monte Carlo simulations in order to calculate the crit-ical Casimir force acting on a colloid in d = B. Scaling functions for the normal critical Casimir force andthe potential
The expressions obtained above within the DA hold forgeneral boundary conditions ( a ) and ( b ) and are valid beyondthe cases we consider in the following, i.e., a ∈ { + , −} and b = − . Figure 2(a) shows the full numerical MFT ( d = K ( ± , − ) with D = com-pared with the corresponding DA results based on the suit-able numerical integration [Eq. (10)] of the analytic (MFT)expression for k ( ± , − ) . Moreover, in Fig. 2, the correspond-ing DA results for d = and by using the corresponding ratio of the correlation lengthsabove and below T c . In Fig. 2(b) we report the difference D J ( Q , D ) : = J (+ , − ) ( Q , D ) − J ( − , − ) ( Q , D ) computed for thevarious cases reported in Fig. 2(a), which will be useful fordescribing the case of a chemically patterned substrate. Thescaling functions in d = D . . d = D the magnitudeof the actual scaling functions becomes larger compared withthose within the DA (corresponding to D →
0) is in agreementwith earlier results obtained for a d -dimensional hypersphere(see, e.g., Ref. 27).It has been shown that the scaling functions obtained withinthe DA for d = corresponding to D . . IV. CHEMICAL STEP ( s ) The basic building block of a chemically patterned sub-strate of the type we consider here, i.e., with translational in-variance in all directions but one ( x ), is a chemical step (s)realized by a substrate with ( a ≷ ) BC for x ≷ ( b ) BC with itscenter located at the lateral position x = X (see Fig. 1 andRef. 32 for experimental realizations). We denote this config-uration by ( a < | a > , b ) . The normal critical Casimir force F s isdescribed by the scaling form F s ( X , D , R , T ) = k B T RD d − K s ( X , Q , D ) , (12)where X = X / √ RD is the scaling variable corresponding tothe lateral position of the colloid. It is useful to write the scal-ing function K s as K s ( X , Q , D ) = K ( a < , b ) + K ( a > , b ) + K ( a < , b ) − K ( a > , b ) y ( a < | a > , b ) ( X , Q , D ) , (13)where the scaling functions of the laterally homogeneous sub-strates K ( a ≷ , b ) depend on Q and D only [Eq. (5)], and the scal-ing function y ( a < | a > , b ) varies from + X → − ¥ to − X → + ¥ , such that the laterally homogeneous cases are recov-ered far from the step. Accordingly, the corresponding criticalCasimir potential F s ( X , D , R , T ) = R ¥ D d z F s ( X , z , R , T ) can becast in the form F s ( X , D , R , T ) = k B T RD d − J s ( X , Q , D ) , (14)and J s ( X , Q , D ) = J ( a < , b ) + J ( a > , b ) + J ( a < , b ) − J ( a > , b ) w ( a < | a > , b ) ( X , Q , D ) , (15)where J ( a ≷ , b ) depend on Q and D only [Eq. (6)], and w ( a < | a > , b ) ( X = ± ¥ , Q , D ) = ∓
1. Note that the scaling func-tions y ( a < | a > , b ) and w ( a < | a > , b ) are independent of the commonprefactor (cid:181) u − [see Sec. II B], which is left undetermined bythe analytical and numerical mean-field calculation of K s and J s . A. Derjaguin approximation
If the sphere is close to the substrate, i.e., D →
0, the DAcan be applied, and one finds for the scaling function of thecritical Casimir force [see Appendix A] y ( a < | a > , b ) ( X ≷ , Q , D → ) = ∓ ± R ¥ + X / d a a − d arccos (cid:0) | X | ( a − ) − / (cid:1) D k ( a Q ) K ( a < , b ) ( Q , D → ) − K ( a > , b ) ( Q , D → ) , (16)where D k ( Q ) = k ( a < , b ) ( Q ) − k ( a > , b ) ( Q ) is the difference be-tween the scaling functions for the critical Casimir forcesacting on two planar walls with ( a < , b ) and with ( a > , b ) boundary conditions, respectively. We note that accordingto Eqs. (16) and (10) within the DA y ( a < | a > , b ) can be de-termined from the knowledge of the film scaling functions k ( a , b ) ( Q ) [Eq. (1)] only. Due to the assumption of additiv-ity which underlies the DA, (i) y ( a < | a > , b ) vanishes at X = Q and it is an antisymmetric function of X and (ii) y ( a < | a > , b ) = y ( a > | a < , b ) ; within the DA both of these proper-ties are valid irrespective of the type of boundary conditionson both sides of the chemical step. (However, the actual scal-ing function y ( a < | a > , b ) as, e.g., obtained from full numericalMFT calculations may violate this symmetry because the ac-tual critical Casimir forces are non-additive.) At the bulk crit-ical point one has Q = y ( a < | a > , b ) ( X , Q = , D → ) = X d − (cid:0) ( − d ) + ( − d ) X − X (cid:1) (cid:0) + X (cid:1) − ( d − ) (17)independent of k ( a ≷ , b ) . Similarly, within the DA one finds forthe scaling function w of the critical Casimir potential [seeAppendix A and Ref. 31] w ( a < | a > , b ) ( X ≷ , Q , D → ) = ∓ ± X R ¥ d s s arccos ( s − / ) −√ s − ( + X s / ) d D k (cid:0) Q [ + X s / ] (cid:1) J ( a < , b ) ( Q , D → ) − J ( a > , b ) ( Q , D → ) . (18)This yields w ( a < | a > , b ) ( X = , Q , D → ) =
0, as expected fromthe underlying assumption of additivity; within full MFT thisonly holds in the limit D →
0. At the critical point we find [seeAppendix A 1] w ( a < | a > , b ) ( X , Q = , D → ) = X (cid:0) − d − X (cid:1)(cid:0) X + (cid:1) − / . (19)For symmetry breaking ( ∓ , − ) BC and Q ≫ f ( ∓ , − ) ( D , T ) acting on two planar walls at a dis-tance D decays (cid:181) exp ( − Q ) [Eqs. (3) and (1)], which withinthe DA leads to the same d-independent result for the scalingfunctions y (+ |− , − ) and w (+ |− , − ) [see Appendix A 2]: y (+ |− , − ) ( X , Q ≫ , D → ) = w (+ |− , − ) ( X , Q ≫ , D → ) = − erf (cid:16)p Q / X (cid:17) , (20)where erf is the error function.Figure 3(a) compares the scaling function w ( a < | a > , b ) for thecritical Casimir potential of a sphere with ( − ) BC in front ofa (+ |− ) step, as obtained within the DA for d = D = /
3. For D . / Q & . The scaling function w ( a < | a > , b ) obtained within the DA ( d =
3) on the basis of the MonteCarlo data of Ref. 23, which is also shown in Fig. 3(a), hasbeen used successfully in order to interpret the experimentaldata of Ref. 32, for which the analysis in terms of separate,independent, and consecutive chemical steps turned out to be X = X / √ RD w ( + | − , − ) ( X , Q , D ) (a) Q = d = d = = . d = d = = . d = d = D → D = / (+) ( − ) (b) K k s ( X , Q , D ) (cid:14) (cid:12)(cid:12) K ( − , − ) ( , ) (cid:12)(cid:12) X = X / √ RD top to bottom: Q = Q = . Q = . Q = . (+) ( − ) DA ( d = n DA ( d = n MFT ( D = ) ( FIG. 3. (a) Scaling function w (+ |− , − ) [Eq. (15)] for the criticalCasimir potential of a spherical colloid with ( − ) BC across a chem-ical step (+ |− ) as a function of X ≡ X / √ RD for various (positive)values of Q = D / x + . Within the DA w (+ |− , − ) is an antisymmetricfunction of X [Eq. (18)] whereas within full MFT this antisymme-try is slightly violated, in particular for small Q . (b) Correspondingscaling function K k s [Eq. (21)] of the lateral critical Casimir force,normalized by the amplitude K ( − , − ) ( , ) = p D ( − , − ) / ( d − ) ofthe normal critical Casimir force at T = T c acting on a colloid with ( − ) BC close to a homogeneous substrate with ( − ) BC within theDA [Sec. III A]. For both (a) and (b) the full numerical MFT re-sults obtained for D = / D → d = and the solid lines refer to d =
4. The lines for Q = Q = . , . , . Q ≫ d . The DA( d =
4) provides a good approximation for the full numerical MFTdata, in particular for Q & K k s > X <
0. Within the DA K k s is a symmetric function of X [Eqs. (18)and (22)] whereas within full MFT this symmetry is slightly violated,in particular for small Q . accurate. Moreover, the critical Casimir forces turned out tobe a sensitive probe of the chemical pattern and its geometricdesign . B. Lateral critical Casimir force
The lateral critical Casimir force is given by F k s = − ¶ X F s and can be cast in the scaling form F k s ( X , D , R , T ) = k B T RD d − (cid:18) DR (cid:19) / K k s ( X , Q , D ) , (21)where K k s is a universal scaling function. F k s and K k s vanishfar from the chemical step, i.e., for | X | → ¥ . In Eq. (21) theprefactors in terms of R and D and their exponents are chosensuch that K k s is regular and non-vanishing for D →
0. We notethat the same holds for the normal critical Casimir forces andthe corresponding potentials [see Eqs. (5), (6), (12), (14), andthe considerations following below].Within the DA K k s can be calculated from Eqs. (15) and(18): K k s ( X , Q , D → ) = − (cid:2) J ( a < , b ) ( Q , D → ) − J ( a > , b ) ( Q , D → ) (cid:3) × ¶ X w ( a < | a > , b ) ( X , Q , D → ) . (22)At bulk criticality Q = K k s ( X , Q = , D → ) = p D k ( ) (cid:0) + X (cid:1) − ( d − ) . (23)For ( ∓ , − ) BC and Q ≫ K k s ( X , Q ≫ , D → ) = (cid:2) J (+ , − ) ( Q , D ) − J ( − , − ) ( Q , D ) (cid:3) r Q p exp (cid:26) − QX (cid:27) , (24)for both d = d =
4. [The prefactor
D J ( Q , D ) = J (+ , − ) ( Q , D ) − J ( − , − ) ( Q , D ) in Eq. (24) is shown in Fig. 2(b).]Figure 3(b) shows the comparison between the normalizedlateral critical Casimir force obtained within the DA (solidlines) and the full MFT data obtained for D = / K k s as a function of X butalso its amplitude is described well by the DA [Eqs. (23) and(24)] for D . /
3, and in particular for Q &
3. We expect thisfeature to hold in d =
3, too, as well as for the normal criticalCasimir force and the critical Casimir potential. The lateralcritical Casimir forces for d = are shown in Fig. 3(b) as dashed lines. Compared with theprevious curves, these ones have similar shapes but their over-all amplitudes in units of the normal critical Casimir force at Q = Q = Q = .
2. Thisdifference reflects the analogous one observed in the normal-ized difference between the corresponding critical Casimir po-tentials for (+ , − ) and ( − , − ) BC, reported in Fig. 2(b).
V. SINGLE CHEMICAL LANE ( ℓ ) In this section we consider the case of a colloid with ( b ) BCclose to a substrate with a single chemical lane ( ℓ ) with ( a ℓ ) BC and width 2 L in the lateral x direction and which is invari-ant along the other lateral direction(s). The remaining parts ofthe substrate are two semi-infinite planes at | x | > L with ( a ) BC [see Fig. 1]. The lateral coordinate X of the center of massof the sphere along the x direction is chosen to vanish in thecenter of the chemical lane. One expects that for “broad” lanesa description in terms of two subsequent chemical steps is ap-propriate [Sec. IV and Ref. 31], whereas for “narrow” lanesthe effects of the two subsequent chemical steps interfere. Wefind that in addition to the variables characterizing the chem-ical step [Eq. (12)], a further scaling variable L = L / √ RD emerges naturally, which corresponds to the width of the lane.Accordingly, the normal critical Casimir force F ℓ acting on thecolloid can be cast in the form F ℓ ( L , X , D , R , T ) = k B T RD d − K ℓ ( L , X , Q , D ) , (25)where K ℓ is the corresponding universal scaling function. Thecritical Casimir potential scales as F ℓ ( L , X , D , R , T ) = k B T RD d − J ℓ ( L , X , Q , D ) , (26)with J ℓ as the universal scaling function for the potential ofa sphere close to a single chemical lane. Analogously toEqs. (13) and (15) we define y ℓ and w ℓ according to K ℓ ( L , X , Q , D ) = K ( a , b ) + K ( a ℓ , b ) + K ( a , b ) − K ( a ℓ , b ) y ℓ ( L , X , Q , D ) , (27)and J ℓ ( L , X , Q , D ) = J ( a , b ) + J ( a ℓ , b ) + J ( a , b ) − J ( a ℓ , b ) w ℓ ( L , X , Q , D ) , (28)so that far from the lane y ℓ ( L , | X | ≫ L , Q , D ) = w ℓ ( L , | X | ≫ L , Q , D ) =
1. On the other hand, onlyfor a “broad” lane the scaling functions at the cen-ter of the chemical lane approach their limiting value y ℓ ( L → ¥ , X = , Q , D ) = − = w ℓ ( L → ¥ , X = , Q , D ) ,corresponding to the homogeneous case with ( a ℓ , b ) BC.
A. Derjaguin approximation
Using the underlying assumption of additivity of the forces,within the DA ( D →
0) we find for the scaling functions of thecritical Casimir force and of the critical Casimir potential [seeAppendix B] y ℓ ( L , X , Q , D → ) = + y ( a ℓ | a , b ) ( X + L , Q , D → ) − y ( a ℓ | a , b ) ( X − L , Q , D → ) (29)and w ℓ ( L , X , Q , D → ) = + w ( a ℓ | a , b ) ( X + L , Q , D → ) − w ( a ℓ | a , b ) ( X − L , Q , D → ) , (30)respectively. Thus, within the DA, from the knowledge of thescaling functions y ( a ℓ | a , b ) [Eq. (16)] and w ( a ℓ | a , b ) [Eq. (18)] forthe chemical step with the appropriate BC, one can directlycalculate the corresponding scaling functions for the chemicallane configuration. Accordingly, in the limit D → y ℓ and w ℓ can be analytically calcu-lated on the basis of Eqs. (29) and (30) by taking advantage ofEqs. (17), (19), and (20). B. Scaling function for the critical Casimir potential
In Fig. 4(a) we show the scaling function w ℓ for the crit-ical Casimir potential obtained within the DA for d = d = T = T c [Eqs. (30) and(19)] for various values of L = L / √ RD as a function of thelateral coordinate of the colloid. One can infer from Fig. 4that, at bulk criticality, the critical Casimir potential variesless pronounced in d = d =
4. As expected, forsmall values of L (i.e., “narrow” chemical lanes), the poten-tial does not reach the limiting homogeneous value − L (i.e., “broad” chemical lanes), w ℓ does attain the value − Q and d . Indeed, from Eqs. (30)and (19) we find that at criticality ( Q =
0) the critical Casimirpotential at the center of the chemical lane ( X =
0) reachesthe limiting value corresponding to the colloid facing a homo-geneous substrate by up to 1% for L & . d = L &
10 in d =
3. We note that the curves in Fig. 4(a) as wellas these bounds are independent of the actual boundary con-ditions because for all kinds of BC the scaling function of thenormal critical Casimir force is constant at the critical point[see Eq. (2)].Below we shall discuss some properties which are specificfor BC with a , a ℓ , b ∈ { + , −} , which exhibit the feature thatthe normal critical Casimir force f ( ∓ , − ) acting on two pla-nar walls decays purely exponentially [see the text precedingEq. (3)] as a function of their distance expressed in units of thebulk correlation length [see Eqs. (1) and (3)]. In Fig. 4(b) thescaling functions w ℓ in d = d = and analytic MFT results , respec-tively, within the DA [see Eqs. (30) and (20)] are shown forthe same values of L as in Fig. 4(a) but off criticality. For Q = . d = d = Q ≫
1, the crit-ical Casimir potential attains its limiting homogeneous valuein the center of the lane for values of L which are smallerthan the ones for Q = d = d = -1-0.500.51 -1 0 1 2 3 4 5 X / L = X / L (a) DA ( D → Q = d = d = w ℓ ( L , X , Q , D → ) = = . = . = . L = . ( a ℓ ) ( a )( a ) ✐✒✑✓✏✒✑✓✏❢❢ -1-0.500.51 -1 0 1 2 3 4 5 X / L = X / L (b) DA ( D → Q = . d = d = w ℓ ( L , X , Q , D → ) = = . = . = . L = . ( − ) (+)(+) FIG. 4. Scaling function w ℓ [Eq. (28)] describing the lateral variationof the critical Casimir potential of a colloid across a single chemicallane of width 2 L as a function of the lateral position X of the col-loid in units of the half width of the lane [see Fig. 1; X = X / √ RD , L = L / √ RD , Q = D / x + ]. Here, w ℓ has been obtained within theDA ( D →
0) in d = Q = Q = . a , a ℓ , b ∈ { + , −} BC [Fig. 1]. For Q ≫ in d = in d = w ℓ = ( a , b ) BC outside the chemical lane, whereas w ℓ = − ( a ℓ , b ) BC as within the chemical lane. Forlarge values of L the critical Casimir potential is the same as for twoindependent chemical steps, and w ℓ reaches its limiting value − X = Q ≫ w ℓ attains − L due to the exponential decay of the critical Casimir force.We note that the DA results for Q = steps for L & . Q = . L & . Q = . w ℓ obtained within the DA( D →
0) at Q = D = /
3. We find a rather good agreement even for small val-ues of L (i.e., “narrow” chemical lanes). This shows that forthe geometry of a colloid close to a single chemical lane, non-linearities, which are actually present in the critical Casimir-1-0.500.51 -1 0 1 2 3 4 5 X / L = X / L d = T = T c D = DA w ℓ ( L , X , Q = , D ) = = . = . = . L = . ( − ) (+)(+) FIG. 5. Test of the performance of the DA for the scaling function w ℓ [Eq. (28)] of the critical Casimir potential for a sphere with ( − ) BCclose to a single chemical lane with ( − ) BC embedded in a substratewith (+)
BC. The MFT w ℓ is evaluated at bulk criticality Q = d = D →
0) and of the full numer-ical MFT (symbols, D = / L = L / √ RD .Nonlinear effects, which are inherently present in the theory, do notstrongly affect the potential. For D → L . effect and potentially invalidate the assumption of additivityunderlying the DA, do not affect the resulting potential forsmall values of D . We expect this property to hold beyondMFT in d = Q = VI. PERIODIC CHEMICAL PATTERNS ( p ) In this section we consider a pattern of chemical stripeswhich are alternating periodically along the x direction. Thepattern consists of stripes of width L with ( a ) BC joinedwith stripes of width L with ( a ) BC, such that the periodic-ity is given by P = L + L . Thus, the geometry of the sub-strate pattern is characterized by the two variables L and P [see Fig. 1]. The coordinate system is chosen such that thelateral coordinate X of the center of the sphere is zero at thecenter of a ( a ) stripe. The normal critical Casimir force F p acting on the colloidal particle and its corresponding potential F p take on the following scaling forms: F p ( L , P , X , D , R , T ) = k B T RD d − K p ( l , P , X , Q , D ) (31)and F p ( L , P , X , D , R , T ) = k B T RD d − J p ( l , P , X , Q , D ) , (32)where P = P / √ RD is the scaling variable characterizing theperiodicity of the pattern and l = L / P is the scaling vari-able chosen to correspond to the relative width of the stripewith ( a ) BC. K p and J p are universal scaling functions forthe normal critical Casimir force and the critical Casimir po-tential, respectively. For l = ( a , b ) BC or ( a , b ) BC, respectively [see Sec. III]. As before it is useful todefine scaling functions y p and w p which vary for l ∈ [ , ] within the range [ − , ] and describe the lateral behavior ofthe critical Casimir effect: K p ( l , P , X , Q , D ) = K ( a , b ) + K ( a , b ) + K ( a , b ) − K ( a , b ) y p ( l , P , X , Q , D ) (33)and J p ( l , P , X , Q , D ) = J ( a , b ) + J ( a , b ) + J ( a , b ) − J ( a , b ) w p ( l , P , X , Q , D ) . (34) A. Derjaguin approximation
Taking advantage of the assumption of additivity of theforces underlying the DA, one finds for the scaling functionof the normal critical Casimir force in the limit D → y p ( l , P , X , Q , D → ) = + ¥ (cid:229) n = − ¥ n y ( a | a , b ) ( X + P ( n + l ) , Q , D → ) − y ( a | a , b ) ( X + P ( n − l ) , Q , D → ) o . (35)Thus, the knowledge of the scaling function y ( a | a , b ) for asingle chemical step with the appropriate BC [Sec. IV] is suf-ficient to calculate directly the corresponding scaling func-tion of the critical Casimir force acting on a colloid closeto a periodic pattern of chemical stripes. As expected, fromEq. (35) one recovers the values y p ( l = , P , X , Q , D ) = y p ( l = , P , X , Q , D ) = −
1, i.e., the cases of a colloidwith ( b ) BC facing a homogeneous substrate with ( a ) BCand ( a ) BC, respectively [see Appendix C].In the limit P →
0, i.e., for a pattern with a very fine struc-ture compared to the size of the colloid, the sum in Eq. (35)turns into an integral [see Appendix C] and, as expected, y p becomes independent of X , i.e., of the lateral position of thecolloid: y p ( l , P → , X , Q , D → ) = − l . (36)Accordingly, in the limit P → K p ( l , P → , X , Q , D → ) = L L + L K ( a , b ) ( Q , D → ) + L L + L K ( a , b ) ( Q , D → ) . (37)For the scaling function of the critical Casimir potential theresults are completely analogous to Eqs. (35)–(37) [see Ap-pendix C].0-1-0.500.51 -0.5 0 0.5 1 1.5 2 X / P = X / P y p ( l , P , X , Q = , D ) (a) l = / T = T c P = P = P = . P = . P = . X / P = X / P y p ( l , P , X , Q = , D ) (b) l = / T = T c P = P = P = . P = . FIG. 6. MFT ( d =
4) scaling function y p [Eq. (33)] of the normalcritical Casimir force acting on a colloidal sphere with ( b ) = ( − ) BC which is close to a periodically patterned substrate [Fig. 1] with ( a ) = ( − ) BC on one kind of stripes [shaded areas] and ( a ) = (+) BC on the other kind of stripes. Due to this choice of the BC thecolloid is attracted by the shaded stripes and repelled by the others. y p is shown as a function of the lateral position of the colloid X / P with P = L + L and at the bulk critical point Q =
0. The geom-etry of the pattern is characterized by P = P / √ RD and l = L / P ,for which we have chosen the values (a) l = . l = . y p as obtained within the DA for d = D = / P .For patterns which are finely structured on the scale of the colloidsize, i.e., P .
2, the actual results deviate from the approximate onesobtained within the DA due to the strong influence (in this context)of the inherent nonlinear effects.
B. Scaling function for the normal critical Casimir force
Figure 6 shows the scaling function y p [Eq. (33)] as a func-tion of X / P = X / P , describing the lateral variation of thenormal critical Casimir force at Q = d = D = /
3; symbols] for symmetry breaking boundary condi-tions ( a ) = ( − ) , ( a ) = (+) , and ( b ) = ( − ) [Fig. 1]. Fromthis comparison for l = . l = . P one can infer that for D → P ≫
1, i.e., L + L ≫ √ RD the DA describes well the ac-tual behavior of the scaling function, even if the force scal-ing function does not attain its limiting homogeneous values y p = ± P . d = T = T c ) the DA does not quantitatively describethe actual behavior and the scaling function y p obtained from -1-0.500.51 -0.5 0 0.5 1 1.5 2 X / P = X / P y p ( l , P , X , Q = , D ) (a) l = / T = T c d = P = P = P = . P = . X / P = X / P y p ( l , P , X , Q = , D ) (b) l = / T = T c P = P = P = . P = . d = n d = n DA, D → FIG. 7. (a) The same as in Fig. 6, but for l = .
8. Also in thiscase, the DA turns out to be accurate for P & P .(b) Comparison between the scaling functions y p in d = d = T = T c , for l = .
8, and within theDA. At the critical point the expression for this scaling function y p is known analytically [see Eqs. (35) and (17)], and the correspondingplot presented here shows that the lateral variation of the normal crit-ical Casimir force is less pronounced in d = d =
4. (We notethat for P → d = P = .
57 in order to show that the critical Casimir forceobtained within the DA practically does not change laterally for suchsmall values of P .) the full numerical MFT calculations deviates from the one ob-tained within the DA. Within both the DA and the full numer-ical MFT calculation, for P → X . But from the full nu-merical calculation we find that the corresponding constantvalue which is attained by y p differs from the one obtainedwithin DA [Eq. (36)]. This shows that for small periodicities P . √ RD nonlinearities inherent in the critical Casimir effectstrongly affect the resulting scaling functions of the force andthe potential, so that in this respect the assumption of additiv-ity of the force and thus the use of the DA are not justified.Figure 7(a) shows the same comparison as Fig. 6 but for l = .
8, which corresponds to an areal occupation of 80% ofthe substrate surface with ( − ) BC and 20% with (+)
BC. Dueto the fact that at the critical point y ( a | a , b ) ( X , Q = , D → ) is actually independent of the BC, y p ( l = . , P , X , Q = , D → ) in Fig. 7(a) is, within the DA, complementary tothe one for l = . y p = X / P of 0 .
5. Instead, the full numerical data in Fig. 7(a) and1-10123 0 1 2 3 v = S − / S + D ph ( S − / S + , S + / L ) . (cid:12)(cid:12) D ( + , + ) (cid:12)(cid:12) S + ( − ) S − (+)(+) L ( v − ) / ( + v ) S + / L = / S + / L = / S + / L = / FIG. 8. Normalized scaling function D ph of the critical Casimirforce at criticality acting on a homogeneous planar wall with (+) BC opposite to a periodically patterned planar substrate with stripesof alternating (+) and ( − ) BC as a function of v = S − / S + , where S + and S − are the respective widths. The symbols correspond to theMFT ( d =
4) data presented in Fig. 12 of Ref. 33 for various valuesof S + / L (note that D ++ = D (+ , +) / ( d − ) in Fig. 12 of Ref. 33). Thedashed and dotted lines which join the data points are a guide to theeye. The solid line corresponds to the DA result given in Eq. (41)which assumes additivity of the forces and turns out to be indepen-dent of the ratio S + / L . One can immediately infer from the graphthat here the assumption of additivity is not justified, which is thelimiting configuration of the sphere-wall geometry for P → Fig. 6(b) show a different behavior as they clearly tend to as-sume the value − ( − , − ) BC. By contrast, for the case l = . + (+ , − ) BC, although the substratearea is covered by 80% with (+)
BC. This feature is addressedin more detail in Sec. VII. Figure 7(b) compares the scalingfunction y p of the normal critical Casimir force at T = T c andfor l = . d = d = T = T c , y p is determined by Eqs. (35) and (17) from which one can in-fer that the lateral variation of the normal Casimir force is lesspronounced for d = d =
4. This qualitative featureholds for all values of l (not shown). However, off critical-ity, Q ≫
1, [according to Eqs. (35) and (20)] the DA scal-ing functions both for d = d = R , the lateral variation of the boundary conditions at thesurface of the patterned substrate on a scale P . √ RD – corre-sponding to the limit P → R does not guaranteethe validity of the DA can be understood by noting that sucha discrepancy between the full numerical calculation and theresult of the DA approximation already emerges in the film ge-ometry (formally corresponding to the limit R → ¥ ), i.e., for a chemically p atterned wall opposite to a laterally h omogeneousflat wall. This “ ph ” configuration has been studied in Ref. 33within MFT for laterally alternating chemical stripes of width L = S + and L = S − with (+) and ( − ) BC, respectively, op-posite to a homogeneous substrate with (+)
BC a distance L apart [see Fig. 1 and the inset of Fig. 8]. Indeed, by using theassumption of additivity of the critical Casimir forces under-lying the DA and neglecting edge effects, the normal criticalCasimir force f ph ( DA ) ( S + , S − , L , T ) per unit area acting on thewalls is predicted to be given by f ph ( DA ) ( S + , S − , L , T ) = S + S + + S − f (+ , +) ( L , T ) + S − S + + S − f (+ , − ) ( L , T ) , (38)where f (+ , ± ) refer to homogeneous parallel walls, as inEq. (1). At the bulk critical point the critical Casimir forceis given in general by f ph ( S + , S − , L , T = T c ) = k B T c d − L d D ph (cid:18) v = S − S + , S + L (cid:19) . (39)Using Eq. (38) together with Eqs. (1) and (2) one finds withinthe DA that ( d − ) D ph ( DA ) (cid:18) v , S + L (cid:19) = v D (+ , − ) + D (+ , +) + v , (40)which renders the rhs of Eq. (40) to be independent of thescaling variable S + / L . Within MFT as studied in Ref. 33 ( d = D (+ , − ) = − D (+ , +) > D ph ( DA ) (cid:18) v , S + L (cid:19) = | D (+ , +) | v − + v . (41)In Fig. 8 we show the comparison between the actual scalingfunction D ph (data points, obtained numerically as reported inFig. 12 of Ref. 33) and D ph ( DA ) (Eq. (41), solid line) derived byassuming additivity of the forces and neglecting edge effects.Figure 8 clearly shows that the actual behavior of the criticalCasimir force in the film geometry is not properly predictedwithin these assumptions. This is expected to be due to thepresence of nonlinear effects and of edge effects in this con-text. This explains why in the limit P → R ≫ D )used here does not capture the behavior of the critical Casimirforce acting on a colloid close to periodically patterned sub-strate.In Fig. 9 we show the behavior of scaling function K p [Eq. (31)] of the normal critical Casimir force acting on thecolloid in d = ( b ) = ( − ) BC as a function of Q = D / x + (i.e., as a function of the normal distance of the colloid fromthe substrate in units of the bulk correlation length) and forvarious values of l and P . In Fig. 9 the scaling function K p isevaluated at X = K p in the whole range of Q for P = .
57 [panel(b)], whereas it does so for P = . P = .
57 the discrepancy between the DA and the numer-ical data is already significant for Q . . . l . . -101234 0 2 4 6 8 10 (a) P = . X = D → D = ) d = (+ , − ) ( − , − ) l = = / = / = / = / l = Q K p ( l , P , X , Q , D ) (cid:14) | K ( − , − ) ( , ) | -101234 0 2 4 6 8 10 (b) P = . X = D → D = ) d = (+ , − ) ( − , − ) l = = / = / = / = / l = Q K p ( l , P , X , Q , D ) (cid:14) | K ( − , − ) ( , ) | -202468 0 2 4 6 8 10 (c) X = d = (+ , − ) ( − , − ) P = . n P = . n l = = / = / = / = / l = Q K p ( l , P , X , Q , D → ) (cid:14) | K ( − , − ) ( , ) | FIG. 9. Scaling function K p [Eq. (31)] of the normal critical Casimirforce acting on a spherical colloid with ( − ) BC located at X = X = X / √ RD ) close to a periodically chemically patterned substrate[see Fig. 1]. K p is suitably normalized by the absolute value of theforce scaling function K ( − , − ) ( , ) = p D ( − , − ) / ( d − ) for the ho-mogeneous ( − , − ) case at criticality and within the DA [Sec. III A].The lateral position of the center of the colloid is fixed at the centerof a stripe with ( a ) = ( − ) BC and width L = l P , which it is at-tracted to, in contrast to the second type of stripes with ( a ) = (+) BC and width L = ( − l ) P , which it is repelled from. The scalingvariable corresponding to the periodicity of the substrate pattern is(a) P = P / √ RD = . P = .
57, whereas the relative areafraction of the ( − ) stripes changes from l = L / ( L + L ) = l = D → d =
4, see Eq. (35)], whereas the symbolsrepresent the full numerical MFT data obtained for D = /
3. TheDA agrees reasonably well with the full data for P = . Q &
1, but for P = .
57 [(b)] it fails to describe the actual behav-ior within the ranges Q . . . l . . K p isshown for P = .
57 and 2 .
3, as obtained for d = .(We note, however, that we do not expect that the curves shown for P = .
57 are quantitatively reliable.) whereas for P = . l ex-cept for Q . P . Q ≫ K p rather well for all valuesof P due to the exponential decay of the critical Casimir forcefor Q ≫ K p for d = . The qualitative features ofthe behavior of K p in d = d = d = P & Q . For smaller val-ues of P , the DA is only quantitatively reliable for large valuesof Q (at which the force decays exponentially). For example,for P & . Q & d = C. Critical Casimir levitation
Rather remarkably, within a certain range of values of l , K p changes sign as a function of Q = D / x + [Fig. 9]. In thiscontext it is convenient to introduce for later purposes anotherscaling variable Y = P | Q | / = P / p R x ± which is indepen-dent of D and therefore does not vanish in the DA limit D ≪ R (i.e., D → K p , there exists acertain value Q = Q ( Y , l , X , D ) at which the normal criticalCasimir force F p acting on the colloid vanishes. This impliesthat in the absence of additional forces the colloid levitates ata height D determined by Q and x + , which can be tunedby changing the temperature. Since for fixed geometrical pa-rameters R , X , and P the scaling variables Q , P , X , and D depend on D , one has to consider the behavior of F p as a func-tion of D near D in order to assess whether the levitationis stable against perturbations of D or not. Stability requires ¶ D F p | D = D < D < D the colloid is repelled fromthe patterned substrate, whereas for for D > D it is attracted).According to Eq. (31) one has ¶ D F p = k B T RD d × {− ( d − ) − P ¶ P − X ¶ X + Q ¶ Q + D ¶ D (cid:9) K p ( l , P , X , Q , D ) . (42)The laterally preferred position is always at X = X =
0, cor-responding to X = X =
0, so that within the DA ( D →
0) onehassign (cid:16) ¶ D F p (cid:12)(cid:12) D = D , X = X , DA (cid:17) = sign (cid:16)(cid:8) − P ¶ P + Q ¶ Q (cid:9) K p ( l , P , X = , Q , D → ) (cid:12)(cid:12) Q = Q (cid:17) , (43)where we have used the implicit equation F p | D = D = K p | D = D =
0. (Equation (43) assumes that ¶ D K p does not di-verge (cid:181) D − for D → Q ≥ ( a ) = ( − ) , ( a ) = (+) , and ( b ) = ( − ) .Within the DA we find that both ¶ P K p | Q = Q , X = X and ¶ Q K p | Q = Q , X = X are negative, so that according to Eq. (43) thesign of ¶ D F p | D = D , X = X , DA can vary and depends on their val-ues as well as on Q and P . However, at criticality ( Q =
0) the30123456789 0 1 2 3 4 5 6 (a) d = Y = P / p R x + Q ( Y , l , X = , D → ) = D / x + l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . . Y ∗ ( l = . ) Y ( l ) Y ( l = ) z }| { (b) d = Y = P / p R x + Q ( Y , l , X = , D → ) = D / x + l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . . Y ∗ ( l = . ) FIG. 10. Values of the scaling variable Q at which within the DA( D →
0) the normal critical Casimir force K p shown in Fig. 9 van-ishes as a function of Y for (a) d = d = and for various values of l = L / P .The solid lines correspond to values of Q for which the levitationof the colloid at a height D above the substrate is stable againstperturbations of D [ ¶ D F p | D = D <
0, see Eq. (43)]. The shaded re-gion and the dashed lines indicate those values of Q for which ¶ D F p | D = D > not correspond to stable levitation. For l > l with l ( d = ) = / l ( d = ) ≃ . Q ceases to ex-ist, i.e., K p does not exhibit a zero. For l < l with l ( d = ) = / l ( d = ) ≃ . Q ( Y ց Y ( l )) diverges. (The values for Y ( l ) are indicated by upward arrows.) For any l < l , Q ex-ists for Y < Y ∗ ( l ) . (From the analysis in Fig. 9 we expect the DAto be quantitatively reliable only for Y & √ Q for Q . Y & . √ Q for Q &
4, which implies l . . d = l . . d = second term of the rhs of Eq. (43) vanishes. Thus, at the bulkcritical point T = T c the derivative ¶ D F p evaluated at D = D and X = X = Q > Q at which the normal criti-cal Casimir force acting on a colloid vanishes as a functionof the new scaling variable Y introduced at the beginningof this subsection, for various l , for X =
0, and within theDA ( D →
0) for (a) d = d =
3. The correspond-ing sign of ¶ D F p (cid:12)(cid:12) D = D [according to Eq. (43)] is also indi-cated: Q drawn as a solid line indicates ¶ D F p (cid:12)(cid:12) D = D < ¶ D F p (cid:12)(cid:12) D = D > D , which occurswithin the shaded regions in Fig. 10. For a given value of l (with l < l < l as we shall discuss in detail further be-low), e.g., l = .
60 in Fig. 10(a), the corresponding curvefor Q shows a bifurcation at Y = Y ∗ ( l ) such that a verticalline drawn in Fig. 10 at a certain Y intersects this curve intwo points Q , u and Q , s > Q , u if Y < Y ∗ ( l ) , whereas it hasno intersection for Y > Y ∗ ( l ) . In the former case Q , u and Q , s correspond to a local maximum and to a local minimumof the critical Casimir potential at distances D , u = x + Q , u and D , s = x + Q , s , respectively, i.e., to an u nstable and a s table levitation point for the colloid, respectively. Instead,for Y > Y ∗ ( l ) , the critical Casimir force has no zero at anyfinite value of D . We note that D = Q = D → ( a ) = ( − ) , ( b ) = ( − ) , and X =
0, see Fig.1] is stabilizedby the steric repulsion of the wall. We note that within theDA the critical Casimir potential for X = (+) BC, i.e., even if 0 = l ≪ X = X =
0, which has been discussed in Sec. V. For given col-loid radius R and width L = l P > L = L / ( √ RD ) diverges as D →
0, sothat the scaling function w ℓ ( L , X , Q , D ) which characterizesthe potential of the lane [see Eq. (28)] attains the value − ( − , − ) BC [see Fig. 4]. Within this approximation and for D ≪ x ± the critical Casimir force becomes attractive if J p ≃ J ℓ < w ℓ ( L , X = , Q → , D → ) < − D (+ , − ) / ( D (+ , − ) − D (+ , +) ) ,i.e., w ℓ < − . d = and w ℓ . − .
76 in d = ; thisoccurs for L > L = . d =
4, and L > L = . d =
3, respectively [see also Fig. 4(a)]. Accordingly, at dis-tances D < l P / ( R L ) (together with D ≪ x ± ) the criti-cal Casimir potential F p is negative and diverges to − ¥ for D →
0. (However, for very small values of l this wouldoccur at distances of microscopic scale such that the scalinglimit and thus the form of F p do no longer hold). Thus thebifurcation of Q at Y ∗ ( l ) corresponds to a transition from(metastable) levitation at D = D , s for Y < Y ∗ ( l ) to stictionat D = Y > Y ∗ ( l ) . For Y < Y ∗ ( l ) the metastable lev-itation minimum at D , s is shielded from the global minimumat D = Y ր Y ∗ ( l ) [see Fig. 11]. Experimentally, one typicallyvaries the value of x + by changing the temperature and4leaves the geometry ( l , P , and R ) unchanged, which resultsin a change of Y via varying T . Thus, experimentally, thetransition at Y ∗ ( l ) corresponds to a de facto irreversible tran-sition from separation to stiction of the colloid as a functionof temperature.Moreover, from Fig. 10 one can infer that for both d = d = l such that, for 1 ≥ l > l , K p has nozero for any choice of Y (i.e., there is no solution Q ) andthe critical Casimir force is attractive at all distances. Withinthe DA, l = D (+ , − ) / ( D (+ , − ) − D ( − , − ) ) [see also Eq. (37)],which renders the values l = .
80 in d = and l ≃ . d = . In addition, from Fig. 10 one can infer that for l > l > l ≃ . Y . Q , s effectively does no longerdepend on Y but solely on l . Accordingly, the distance D , s (cid:181) x + at which the colloid stably levitates can be tunedby temperature upon approaching criticality. However, for l < l ≃ . Q , s diverges at Y = Y ( l ) < Y ∗ ( l ) such thatfor Y ( l ) < Y < Y ∗ ( l ) the colloid exhibits critical Casimirlevitation at a local minimum of the potential, whereas withinthis range of l values for Y < Y ( l ) the critical Casimirpotential has only a local (positive) maximum at D , u ; it isrepulsive for D > D , u and therefore for large values of D (i.e., Q ≫ P ≪
1) it approaches zero from positive val-ues. This qualitative change in the behavior of the criticalCasimir potential occurs at l = l . The value of l is closeto 0.5 because the repulsive and attractive forces for (+ , − ) and ( − , − ) BC, respectively, have similar strengths but oppo-site signs for Q ≫
1, i.e., k (+ , − ) ( Q ≫ ) ≃ − k ( − , − ) ( Q ≫ ) for both d = d = | A − / A + | ≃ . d = and | A − / A + | = d = ]. Accordingly, de-pending on l being larger or smaller than l ≃ .
5, the areacovered by one of the two BC prevails and the resulting forceis asymptotically (i.e., Q ≫
1) attractive or repulsive, respec-tively [see the remark at the end of Sec. VI A and Eqs. (37)and (A29)]. Taking into account the slight difference in thestrength of the asymptotic forces for (+ , − ) and ( − , − ) BCone finds l = ( − A + / A − ) − which renders l = / d = l ≃ .
545 in d =
3. The asymptotic behavior ofthe force at large distances can be inferred from the asymp-totic behavior of K p ( l , P = YQ − / , X = , Q ≫ , D → ) ≃ A ( Y , l ) Q d − e − Q , which can be obtained from Eqs. (35),(33), (A29), (20), and (3) . Accordingly, the value Y ( l ) atwhich Q , s diverges is characterized by the fact that A ( Y ≶ Y ( l ) , l ) ≷ Y < Y ( l ) or Y > Y ( l ) ,respectively. The condition A ( Y ( l ) , l ) = Y ( l ) :2 l = ¥ (cid:229) n = − ¥ erf n Y ( l ) √ ( n + l ) o − erf n Y ( l ) √ ( n − l ) o . (44)For l ≪ n = Y ( l ≪ ) ≃ / l − erf − ( l ) , where erf − is the inverse error function,which yields the relations Y ( l ≪ ) ≃ . / l for d = Y ( l ≪ ) ≃ . / l for d =
4. On the other hand, inthe marginal case one expects Y ( l = l ) =
0. However, asargued above, at the critical point ( Q =
0) the colloid doesnot exhibit stable levitation for any geometrical configura-tion; this is in accordance with Fig. 10 because for T → T c ,the levitation minimum of the potential moves to large D ( D , s = Q , s x + → ¥ ) and disappears at T = T c . In summary, as function of l there are three distinct levita-tion regimes:(i) l > l with l ( d = ) ≃ .
88 and l ( d = ) = / l > l > l with l ( d = ) ≃ .
545 and l ( d = ) = /
2: Sufficiently close to T c , i.e., for Y = P / p R x + < Y ∗ ( l ) there is a local critical Casimir levitation mini-mum. Upon approaching T c its position D , s = Q , s x + ,with Q , s ( x + → ¥ ) finite, moves to macroscopic valuesproportional to the bulk correlation length.(iii) l > l : As in (ii) there is a local critical Casimirlevitation minimum sufficiently close to T c , i.e., for Y < Y ∗ ( l ) . In general the onset of its appearanceoccurs further away from T c upon lowering l . Uponapproaching T c the position D , s of this minimum di-verges at a distinct nonzero reduced temperature givenby Y ( l ) , i.e., at x + = P / [ R Y ( l )] : D , s = Q , s x + with Q , s ( Y ց Y ( l )) → ¥ .We note that, according to Figs. 6, 7, 8 and 9, we expectthat for P . Q . P . . Q &
4, theDA does not provide a quantitatively reliable description ofthe actual behavior of K p and therefore of F p ; thus, for valuesof Y . √ Q for Q .
4, and Y . . √ Q for Q &
4, weexpect quantitative discrepancies between the actual behaviorand the one predicted by the DA shown in Fig. 10. Nonethe-less our results demonstrate that the geometric arrangement ofthe chemical patterns allows one to design the normal criticalCasimir force over a wide range.Figures 11(a) and (b) show the critical Casimir potential F p as a function of D in d = for a variety ofspecifically chosen values of the parameters P , L , R , and x .The choice of these values is motivated by the typical exper-imental parameters which characterize recent investigationsof the critical Casimir force acting on colloids immersed inbinary liquid mixtures . In particular, concerning thecolloid radius we focus on the data of Ref. 53, correspond-ing to R = . m m, while for the pattern we have chosena periodicity P = m m with l = . L = L = P = . m m with l = .
65 (i.e., L = L = .[We note that F p as shown in Fig. 11(a) and (b) is expected todescribe the actual interaction potential in the scaling regimecharacterized by values of D and x + much larger than mi-croscopic length scales (such as x + ≃ . ) so that thisprediction for F p is valid only for D , x + & F p for various val-ues of x + within an experimentally accessible range .From Figs. 11(a) and 11(b) one can infer that for small valuesof x + (corresponding to large values of Y > Y ∗ ( l ) ) the crit-ical Casimir potential is always attractive with a monotonicdependence on D [see also Fig. 10]. Upon approaching criti-cality, i.e., for increasing values of x + and decreasing valuesof Y < Y ∗ ( l ) , a local maximum and a local minimum of thepotential develop, so that for very small as well as for large5 -15-10-5051015 0 50 100 150 200 D [nm] F p ( L , P , X = , D , R , T ) / k B T (a) R = . m m, P = m m l = . = = = = = = x + = Y = = = = = = = x + = Y = D [nm] F p ( L , P , X = , D , R , T ) / k B T (b) R = . m m P = . m m, l = . = = = = = = = = = = x + = Y = = = x + = Y = D [nm] [ F p ( L , P , X = , D , R , T ) + F e l ( D ) ] / k B T (c) F el ( D ) / k B T = e − ( D − ) / R = . m m, P = . m m, l = . x + = Y = = = = = = = = = x + = Y = = = = = M i n i m a f o r l = FIG. 11. Critical Casimir potential F p [Eq. (32)] in d = R = . m m close to a periodically patterned substrateas a function of D and for various values of x + for P = m m with l = . P = . m m with l = .
65 in (b) and (c). Thevalues of P , l , and x + are chosen as to be experimentally accessiblein a colloidal suspension exhibiting critical Casimir forces .The critical Casimir potential for the colloid close to a patterned sub-strate may exhibit – depending on the value of x + , and, thus, on thetemperature – a local minimum corresponding to stable levitation.In (c) an electrostatic potential F el [Eq. (45)] is added to F p , whichrefers to actual experimental data . The shaded area indicates theranges of the positions and the depths of the local minima of the to-tal potential occurring if the substrate is laterally homogeneous andpurely attractive, i.e., for l = ( − , − ) BC) for the range 14nm < x + < . k B T and 70 k B T (indicated by the shaded arrow); for l = D , s = Q , s x + due to critical Casimir levitation can be much larger and tuned bytemperature. Moreover, whereas for l = T c the minima monotonically become deeper, the levitation minima firstdeepen and move to smaller values of D followed by a decrease ofthe depth, by becoming more shallow, and moving to larger values of D . Reducing the range and strength of the electrostatic repulsion byadding salt to the solvent is expected to provide access to even moredetails of the critical Casimir levitation potential F p shown in (b). D the colloid is attracted to the patterned substrate, whereaswithin an intermediate range of values for D it is repelledfrom it [see also Fig. 10]. Thus, the colloid stably levitatesat a distance D , s corresponding to a local minimum of thepotential. The depth of this minimum ranges between a few k B T [Fig. 11(a)] up to several k B T [Fig. 11(b)]. Upon in-creasing x + , D , s increases as well, i.e., the colloid positionis shifted away from the patterned substrate with the poten-tial minimum becoming more shallow. In Fig. 11(a) l = . Y ∗ ( l = . ) ≃ .
65 and Y ( l = . ) ≃ .
71 [seeFig. 10(b)] so that for Y < Y ( l = . ) , i.e., for x + & . D uponapproaching the temperature corresponding to x + ≃ . l = .
65 and one has Y ∗ ( l = . ) ≃ .
63; here Q , s remains finite for Y → l < . x + > P / [ R ( Y ∗ ( l = . )) ] ≃ T c the levitation minimum moves tomacroscopic values of D proportional to the bulk correlationlength x + .The discussion above focuses on the position of mechani-cal equilibrium of the colloid, corresponding to the point atwhich the forces acting on the particle vanish and the as-sociated potential F has a local minimum F min . However,due to the thermal fluctuations of the surrounding near-criticalfluid at temperature T , the colloid undergoes a Brownian dif-fusion which allows it to explore randomly such regions inspace where the potential F is typically larger than F min for at most few k B T . As a result, a position of mechani-cal equilibrium is stable against the effect of thermal fluctua-tions only if the potential depth of the minimum is larger thanfew k B T . In particular, if the potential barrier F ( L , P , , D = D , u , R , T ) − F ( L , P , , D = D , s , R , T ) , which separates theposition of the local minimum at distance D = D , s (levita-tion) from the global one at D = x + = x + . [see also Sec. VIII below]. Concern-ing the spatial dependence of the electrostatic repulsion weconsider the one of Ref. 53, which corresponds to a colloid ofradius R = . m m immersed in a near-critical water-lutidinemixture and close to a substrate exhibiting critical adsorptionof water or lutidine : F el ( D ) / k B T = exp {− k ( D − D ) } , (45)where D = k − = . (Formally, F el inEq. (45) is finite for D →
0, and thus F p + F el is negativefor D . D = F p → − ¥ for D →
0. However, Eq. (45) is actually theasymptotic form of the electrostatic interaction which is valid6for distances larger than the electrostatic screening length, i.e., D ≫ k − . The corresponding total potential F p + F el is there-fore not accurate for small values of D and is reported inFig. 11(c) for D > P = . m m, l = .
65, and experimentally accessible valuesof x + . Figure 11(c) provides a realistic comparison of thecritical Casimir potential with other forces as they typicallyoccur in actual experimental systems. One can infer fromthe graph reported in Fig. 11(c) that for this choice of pa-rameters the critical Casimir levitation exhibited by the col-loid is rather pronounced even in the presence of electrostaticinteraction. Far from the critical point ( x + = . x + . x + the local minimum of the crit-ical Casimir potential corresponding to levitation is locatedat distances D , s . x + & D , s & x + , which allows for measurements of the critical Casimirpotential for distances at which the precise form of F el is notimportant. Moreover, the depth of the minimum decreases upon approaching criticality and the minimum becomes moreshallow. This behavior of the levitation minimum is distinctfrom the critical Casimir effect acting on a colloid close toa homogeneous substrate: a local minimum also occurs inthe latter case if the critical Casimir force is purely attrac-tive ( l = ( − , − ) BC) and works against the electrostaticrepulsion , due to the competition of different forces withopposite sign. (We note that the critical Casimir levitation de-scribed above emerges from the critical Casimir force alone,i.e., it is a feature of a single force contribution.) However, inthis homogeneous case the preferred colloid position D , ( − , − ) depends crucially on the form of the electrostatic interactionand is almost constant (50nm < D , ( − , − ) < x + and become much larger than those shownin Fig. 11(c) (see, e.g., Fig. 2(a) and Fig. 2(c) in Ref. 6 andFig. 3 in Ref. 53). In Fig. 11(c) this is indicated by the shadedarea and the shaded arrow, which corresponds to the area ofthe graph within which minima of the total potential in thehomogeneous case l = < x + < . k B T up to 70 k B T . On theother hand, the colloid position D , s due to critical Casimirlevitation can be much larger, can reach values of several x + ,and can be tuned by temperature according to D , s = Q , s x + .In conclusion, the examples presented in Fig. 11 strongly sug-gest that the critical Casimir levitation of a colloid close to apatterned substrate is experimentally accessible.By patterning the substrate, one introduces an additional( lateral ) length scale into the system, which, according toour results presented above, can finally lead to stable levita-tion. Introducing an additional length scale along the nor- mal direction by stacking different materials on top of eachother may lead to levitation due to quantum-electrodynamic Casimir forces . The behavior of the stable levitation dis-tance shows a bifurcation and irreversible transitions fromseparation to stiction similarly to the ones described above[see Fig. 10]. In that context great importance has been givento the temperature dependence of the position D , s of stablequantum Casimir levitation , which is quantified by the valueof ddT D , s . In the critical Casimir case presented here, for anestimate of ddT D , s we pick as an example the stable levitationpositions for x + = x + = x + = . T c ≃ .Therefore, according to x + / x + = | ( T − T c ) / T c | − n , the dif-ference in temperature required to move from x + = x + = D T ≃ . ddT D , s ≃ − for the average temperature dependence of critical Casimirlevitation [Fig. 11(b)], and ddT D , s ≃ − by addition-ally taking electrostatics into account [Fig. 11(c)]. We notethat in the present critical case ddT D , s can become arbitrar-ily large at temperatures corresponding to the transition fromseparation to stiction and the emergence of the local mini-mum and the local maximum of the critical Casimir potential[see Fig. 10 and the curves for x + = x + = ddT D , s being two ordersof magnitude larger than the one predicted for the quantum-electrodynamic Casimir effect in Ref. 56. In general the col-loid will not only be exposed to the critical Casimir force andto an electrostatic force but also to gravity and to laser tweez-ers, which generate a linearly increasing potential contribu-tion. This attractive contribution tends to reduce the potentialbarriers shown in Fig. 11 and can eliminate small barriers al-together. Thus these external forces can be used to switchlevitation on and off (compare a similar discussion related tothe quantum-electrodynamic Casimir levitation in Ref. 56).
VII. CYLINDER
Currently, there is an increasing experimental interest in elongated colloidal particles which have a typical diameterof up to several 100 nm and a much larger length (see, e.g.,Refs. 30 and 57 and references therein). These types of col-loids resemble cylinders rather than spheres. The descriptionof their behavior in confined critical solvents calls for a natu-ral extension of the studies presented in Secs. III–VI. Hence,in the present section we consider the case of a 3 d cylinderwith ( − ) BC which is adjacent and parallel aligned to a pe-riodically chemically patterned substrate consisting of alter-nating ( − ) and (+) stripes as the ones discussed in Sec. VI.Accordingly, the axis of rotational invariance of the cylinderis perpendicular to both the x direction [Fig. 1] and the di-rection normal to the substrate, and it is parallel to the direc-tion of spatial translational invariance of the chemical stripesforming the pattern. As compared with the case of the spherethe analysis for the cylinder is technically simpler because thesystem as a whole is invariant along all directions but two,the lateral one, x , and the one normal to the substrate. (For7the sphere its finite extension in the second lateral direction,which is normal to the x -axis, matters and thus leads to a ba-sically three-dimensional problem. Accordingly, here we donot consider short cylinders, for which this finite length mat-ters, too.) This reduction of the number of relevant dimen-sions allows us to perform numerical calculations of adequateprecision for a range of various pattern geometries which iswider than in the case of the sphere. (Here, we do not con-sider a cylindrical colloid which is not perfectly aligned withthe pattern and which would, therefore, experience a criticalCasimir torque .) Even though the expressions derived inAppendix D can be used to study the case of a cylinder hav-ing its axis laterally displaced by an arbitrary amount X fromthe chemical step, our numerical calculations for the case of achemical stripe address only the case X =
0. This correspondsto a lateral position of the symmetry axis of the cylinder whichcoincides with the center of an attractive ( − ) stripe.In Appendix D we briefly derive the scaling behavior ofthe normal critical Casimir force acting on the cylinder andcompare it with the case of a sphere. Then, we adapt theDerjaguin approximation appropriate for the geometry of thecylinder. On this basis, we have calculated the scaling func-tion of the normal critical Casimir force acting on the cylinderin d = d = and of the analytic MFT expres-sion for the critical Casimir force for the film geometry , re-spectively. In addition, within the same approach as the oneof Sec. II B we have calculated numerically the MFT scalingfunctions corresponding to D =
0, in order to assess the per-formance of the DA.Here we focus on the comparison between the DA appro-priate for the cylinder and the full numerical MFT data for thescaling function K cylp ( l , P , X = , Q , D ) which characterizesthe normal critical Casimir force in the presence of a peri-odically patterned substrate; l , P , X , Q , and D are definedas in the case of the sphere [see Sec. VI and Appendix D].Figure 12 shows the scaling function of the normal criticalCasimir force acting on a cylinder as a function of Q as ob-tained from the DA ( D →
0) in d = D = /
3. Besides the quanti-tative differences in the scaling function as a function of Q ,the qualitative features of the behavior of the force acting ona cylinder, which is reported in Fig. 12 for various values of l , are similar to the ones for the sphere [compare Fig. 9].For P = .
92 [Fig. 12(a)] the DA describes the actual behav-ior of the critical Casimir force rather well, in particular for Q &
2, even for most values of l . As in Fig. 9, for a certainrange of values of l the normal critical Casimir force changessign at Q cyl0 ( P , l , X = , D ) . On the other hand for small pe-riodicities ( P = .
29 in Fig. 12(b)) the DA in d = l & . K cylp of the normal crit-ical Casimir force obtained numerically and represented bysymbols in Fig. 12(b) is very close (much closer than withinthe DA) to the one corresponding to the homogeneous casewith ( − , − ) BC (corresponding to l =
1) and does not showa change of sign. This means that, even if the substrate is -101234 0 2 4 6 8 10lines: DA ( D → D = ) P = . (a) X = d = l = = / = / = / = / l = Q K c y l p ( l , P , X , Q , D ) (cid:14) | K c y l ( − , − ) ( , ) | -101234 0 2 4 6 8 10lines: DA ( D → D = ) P = . (b) X = d = l = = / = / = / = / l = Q K c y l p ( l , P , X , Q , D ) (cid:14) | K c y l ( − , − ) ( , ) | -202468 0 2 4 6 8 10 (c) X = d = (+ , − ) ( − , − ) P = . n P = . n l = = / = / = / = / l = Q K c y l p ( l , P , X , Q , D → ) (cid:14) | K c y l ( − , − ) ( , ) | FIG. 12. Normalized scaling function K cylp [see Appendix D, includ-ing expressions for K cyl ( − , − ) ( , ) ] of the normal critical Casimir forceacting on a cylindrical colloid close to and parallel to a periodicallypatterned substrate. The cylinder axis is aligned with the striped pat-tern and positioned above the center of a ( − ) stripe which has thesame adsorption preference as the cylinder (analogous to Fig. 9 for aspherical colloid). In (a) for P = .
92 the appropriate DA describesthe actual MFT data rather well, and for 0 . . l . . l = l =
1, for P = .
29 the DA fails todescribe quantitatively the actual behavior [see the main text]. In (c) K cylp is shown for d = for the two cases P = . P = .
92. We expect that also in d = P = .
29 isnot quantitatively reliable. not homogeneous but chemically patterned – but such that thelarger part of the surface still corresponds to ( − ) BC, i.e., l & . ( − ) BC resembles the behavior for laterally ho-mogeneous ( − , − ) BC. This can be understood in terms ofthe fixed point Hamiltonian in Eq. (4) which penalizes spa-tial variations of the order parameter at short scales. Thus8the system tries to smooth out spatial inhomogeneities of theorder parameter profile, biased by the preference of the col-loidal particle. If the pattern is very finely structured, i.e., P = ( L + L ) / √ RD ≪
1, regions with a positive order pa-rameter close to the narrow (+) stripes ( l ≃ − , i.e., L ≪ L )extent only very little into the direction normal to the substrateand the resulting order parameter profile at a distance fromthe substrate remains negative only , so that the force resem-bles the one corresponding to the homogeneous case. (Notethat within the DA, the corresponding order parameter profilewould simply consist of a patchwork of the order parameterprofiles corresponding to the film geometry, with no smooth-ing taking place at the edges of the various spatial regions.)Similarly, but in a weaker manner due to the opposite orderparameter preference at the colloid, the curves in Fig. 12(b)for l . . (+ , − ) (i.e., l = l = / l = l = / l = (+ , − ) boundary condi-tions is energetically less preferred than the one with ( − , − ) boundary conditions because in the (+ , − ) case an interfaceemerges between the two phases. For broad stripes, i.e., incontrast to the case P →
0, the energy costs for a similar be-havior are seemingly larger: the full numerical MFT data for l = / l = / l = l =
1, respectively, for P = .
92 than for P = . K cylp of the normalcritical Casimir force for d = .One can infer from Fig. 12(c) that the qualitative featuresof the MFT scaling function as described above, such as thechange of sign, are carried over to d = D from the substrate only if ¶ D F cylp | D = D <
0. Within the DA and at the laterally stableposition X = ¶ D F cylp is given by Eq. (43) with K p replaced by K cylp . The behavior of Q cyl0 as a function of Y and the demarcation of the regions where levitation is sta-ble against perturbations of D is shown in Fig. 13, where thesolid and the dashed lines correspond to stable and unstablelevitation, respectively. The behavior for the normal criticalCasimir force acting on the cylinder is qualitatively similar tothe one for the sphere shown in Fig. 10. Analogously to thecase of a sphere discussed in Sec. VI C, no stable levitationis found at T = T c or for l > l = D (+ , − ) / ( D (+ , − ) − D ( − , − ) ) ,where l = .
80 in d = l ≃ .
88 in d =
3. On theother hand, for Q >
0, and l < l , it is always possible tofind values of P and R such that stable levitation of the cylin-der occurs at a certain distance from the substrate. The val-ues of l below which one has a finite value Y ( l ) at which Q diverges remain the same as for the case of a sphere, i.e., l ( d = ) = / l ( d = ) ≃ . Y ( l ) remain the same [see Eq. (44)]. (a) d = Y Q c y l ( Y , l , X = , D → ) l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . Y ( l ) Y ( l = ) z }| { Y ∗ ( l = . ) (b) d = Y = P / p R x + Q c y l ( Y , l , X = , D → ) l = . l = . l = . l = . l = . l = . l = . l = . l = . l = . Y ∗ ( l = . ) l = . l = . FIG. 13. Values of the scaling variable Q cyl0 at which the normal crit-ical Casimir force K cylp acting on a cylinder close to a periodicallypatterned substrate vanishes as a function of Y = P / p R x + [com-pare Fig. 10 for the case of a sphere] within the DA. The regionindicated by solid lines corresponds to the one in which the levi-tation of the cylinder at a height D = D = Q x + is stable againstsmall perturbations of D , wheres in the shaded region indicated bydashed lines there is no such stable levitation although the normalcritical Casimir force acting on the colloid vanishes. For l > l with l ( d = ) = / l ( d = ) ≃ . Q cyl0 ceases to exist,i.e., K cylp does not exhibit a zero. For l < l with l ( d = ) = / l ( d = ) ≃ . Q cyl0 ( Y ց Y ( l )) diverges. (The values for Y ( l ) are indicated by upward arrows.) For any l < l , Q cyl0 existsfor Y < Y ∗ ( l ) . We expect the DA to be quantitatively reliable onlyfor Y / √ Q & Q . Y / √ Q & . Q & VIII. SUMMARY AND CONCLUSIONS
We have investigated the universal properties of the nor-mal and lateral critical Casimir forces acting on a spherical orcylindrical colloidal particle close to a chemically structuredsubstrate with laterally varying adsorption preferences for thespecies of a (near) critical classical binary liquid mixture (atits critical composition) in which the colloid is immersed.Within the Derjaguin approximation (DA) [see Fig. 14] in spa-tial dimensions d = d = T = T c and – for symmetry breaking boundaryconditions – far away from the critical point. These relationsenable one to obtain predictions for actual three-dimensionalsystems with a sphere-inhomogeneous plate geometry (forwhich currently computations are not possible) based on thescaling function for the parallel homogeneous plate geometry,for which, e.g., Monte Carlo simulation data in d = d =
4) and symmetry-breaking boundary con-ditions [Sec. II B] have been obtained fully numerically andhave been compared with the approximate results of the DA,which allows us to explore the limits of validity of the latter.We have studied several relevant situations [see Fig. 1] andour main findings are the following:1. First, we have studied a spherical colloid immersed ina binary liquid mixture close to a chemically homoge-neous substrate which has, compared to the colloid, thesame ( − ) or a different (+) adsorption preference forone of the species of the mixture [Sec. III]. Close tothe bulk critical point at T = T c the critical Casimirforce induced by the confinement of the order param-eter (e.g., the concentration difference in a binary liquidmixture) can be described in terms of universal scalingfunctions depending on the surface-to-surface distance D of the colloid from the substrate scaled by the bulkcorrelation length, Q = sign (( T − T c ) / T c ) D / x ± , and itsratio with the radius of the colloid, D = D / R [Eqs. (5)and (6)]. The scaling functions obtained within the DA[Eqs. (10) and (11)] are valid for D →
0. From the com-parison with the full numerical MFT results [Fig. 2] wefind that in d = D . .
4. Based on Monte Carlo simula-tion data for the scaling function of the critical Casimirforce between parallel, homogeneous plates and withinthe DA we have obtained also the scaling function forthe critical Casimir force on a spherical colloid close toa homogeneous substrate in d = chemical step , which we have studied inSec. IV. Due to the broken translational invariance inone lateral direction ( x ) the critical Casimir forces andpotentials acquire a dependence on the additional scal-ing variable X = X / √ RD , which corresponds to the lat-eral distance X of the center of the spherical colloidfrom the position of the chemical step along the plane[Eqs. (12), (14), and (21)]. Due to the different bound-ary conditions on both sides of the chemical step a lat-eral critical Casimir force emerges, which leads to alaterally varying potential for the colloid. In the limit D → d = , which revealed that the criticalCasimir effect is rather sensitive to the geometrical de-tails of the substrate patterns.3. Section V deals with the critical Casimir forces andthe corresponding potential acting on a spherical col-loid in front of a single chemical lane of width 2 L ,which additionally depends on a fourth scaling vari-able L = L / √ RD [Eqs. (25) and (26)]. It turns outthat within the DA the scaling functions for the criticalCasimir force and the critical Casimir potential across achemical lane can be expressed in terms of the ones forthe chemical step [Eqs. (29) and (30)]. For large valuesof L the resulting potential can be described as a suit-able superposition of chemical steps, whereas for L . D . .
4, even for small L . Seemingly,in this respect, the nonlinearities inherent in the criti-cal Casimir effect and edge effects do not considerablyaffect the resulting scaling functions [Fig. 5].4. On the basis of the results of Sec. V, in Sec. VI wehave studied the universal scaling functions of the crit-ical Casimir force and the corresponding potential fora sphere opposite to a periodically patterned substrate with laterally alternating chemical stripes of differentadsorption preferences [Sec. VI]. These scaling func-tions [Eqs. (31) and (32)] depend, besides the scalingvariables Q , D , and X , on two additional scaling vari-ables P = P / √ RD and l = L / P , which correspond tothe period P = L + L of the pattern and to the width L ≤ P of the stripes with the same adsorption prefer-ence as the colloid. The scaling function for the normalcritical Casimir force obtained within the DA can be ex-pressed in terms of the one for the chemical step and de-scribes the actual behavior well for P & P → P → film geometryof a patterned wall next to a laterally homogeneous flatwall, additivity of the critical Casimir forces does nothold [Fig. 8].5. The MFT scaling function of the normal criticalCasimir force acting on a colloid close to a periodi-cally patterned substrate shows a remarkable behavioras a function of Q = D / x + . Within a certain range ofvalues of P and l the critical Casimir force vanishesat Q corresponding to a distance D = D between thecolloid and the substrate. We have analyzed the signof the derivative of the critical Casimir force with re-spect to D at D , which is negative if for D < D = D , s the colloid is repelled from the substrate whereas for0 D > D = D , s it is attracted to the substrate [Fig. 10].This means that in the absence of other forces the col-loid can levitate above the substrate at a stable distancewhich can be tuned by temperature. Stable levitationpoints are found also in d =
3, within the DA and on thebasis of the Monte Carlo data for the parallel plate ge-ometry [Figs. 7(b), 9(c), and 10(b)]. Our analysis showsthat at the critical point T = T c levitation is not possi-ble, whereas off criticality a geometrical configurationleading to stable levitation can always be found. Forfixed geometrical parameters, the critical Casimir po-tential as a function of D changes from a monotonic be-havior to a non-monotonic one upon approaching criti-cality; a local maximum and a local minimum, the lattercorresponding to stable levitation, occur [Fig. 11(a) and(b)]. Experimentally, this corresponds to a de facto ir-reversible transition from separation to stiction of a col-loid and a patterned substrate. The depths of these po-tential minima can be up to several k B T so that the levi-tation is stable against Brownian motion of the colloid.The critical Casimir levitation can be rather pronouncedand robust even in the presence of electrostatic interac-tions [Fig. 11(c)]. The levitation height is proportionalto the bulk correlation length and thus can be tunedby varying temperature. Depending on the geometricparameter l we have identified two distinct types oftemperature dependences of the levitation height D , s .In both cases it exhibits a high temperature sensitivity ddT D , s which, for realistic examples at room tempera-ture, is of the order of several 100nm K − . These re-sults show that the periodic patterning of the substrateenables one to design critical Casimir forces over a widerange of properties.6. This behavior is also observed for a cylindrical colloidwhich lies parallel to the substrate such that its axis isaligned with the translationally invariant direction of thestripes [Sec. VII and Appendix D]. The main features ofthe scaling function for the corresponding normal criti-cal Casimir force are similar to the ones for the spheri-cal colloid: the DA describes well the actual behavior asobtained from full numerical MFT calculations for largevalues of P , but fails quantitatively for P . P → d = [see Fig. 11(c)]. Upon approaching the critical point inthe phase diagram, experiments and theory (see, e.g., Refs. 7, 31, and 33) highlight the importance and the rele-vance of the critical Casimir effect in comparison with theseother forces.The lateral critical Casimir forces occurring for patternedsubstrates as discussed here are highly sensitive to the detailsof the geometry of the pattern. A detailed comparison withavailable experimental data has to take this into account .This sensitivity even allows for an independent determina-tion of the geometry of a chemically structured substrate bymeans of the critical Casimir effect. This is useful in casesin which it is difficult to infer the geometry of the chemicalpattern directly . Concerning the comparison with experi-ments for chemically structured substrates, the theoretical pre-dictions for the critical Casimir force are in agreement withthe presently available data , for which the description interms of independent chemical steps [Sec. IV] turns out to besufficient . In order to test our specific predictions obtainedfor narrow single chemical lanes and for periodic chemicalstripes, structures on the nanometer scale are needed. Prelim-inary experimental data in this direction are encouraging .In view of present basic research efforts and potential ap-plications, it is important to study the effect of weak criticaladsorption of the fluid at the confining surfaces, correspond-ing to finite surface fields. Such weak surface fields can berealized by applying suitable surface chemistry and they in-fluence the resulting behavior of the critical Casimir effectstrongly . Another approach to create an effective reduc-tion of the surface adsorption is to create fine periodic chem-ical patterns with different (strong) adsorption preferences asdiscussed here. However, our results for P → short dis-tances because in this range the critical Casimir force for ainhomogeneous adsorption preference resembles the one fora homogeneous substrate corresponding to strong adsorption.On the other hand, at large distances a periodically patternedsubstrate does lead to an effective BC corresponding to a weakadsorption preference, and for l = / . This offers the interesting perspective to study,at least asymptotically, critical Casimir forces with Dirich-let BC by using classical fluids instead of superfluid quantumfluids .A patterning on the molecular scale is not captured by thecontinuous approach pursued here, which gives the universalfeatures of the critical Casimir effect. Nonetheless, a molecu-lar patterning of the substrates may provide another means foran effective reduction of the adsorption of the correspondingfluid at the surface. However, on a molecular scale the pat-terning is more likely to lead to randomly distributed surfacefields which opens a new challenge in the context of criticalCasimir forces. ACKNOWLEDGMENTS
S. K. and L. H. gratefully acknowledge support by grantHA 2935/4-1 of the Deutsche Forschungsgemeinschaft. A. G.is supported by MIUR within the program “Incentivazionealla mobilità di studiosi stranieri e italiani residenti all’estero”.1
Appendix A: Derjaguin approximation for a chemical step
In this appendix we first calculate within the DA the nor-mal critical Casimir force F s ( X , D , R , T ) [Eq. (12)] actingon a spherical colloid of radius R facing a chemical stepby using the DA. (We cannot directly calculate the lateralcritical Casimir force F k s ( X , D , R , T ) within the DA becausefor two parallel homogeneous plates such a force vanishes.)In a second step we derive the critical Casimir potential F s ( X , D , R , T ) = R ¥ D d z F s ( X , z , R , T ) by integrating this re-sult for the normal critical Casimir force. In a third step thelateral critical Casimir force is obtained as F k s ( X , D , R , T ) = − ¶ X F s ( X , D , R , T ) = − R ¥ D d z ¶ X F s ( X , z , R , T ) [see Sec. IV B].In the spirit of the DA, the surface of the spherical col-loid with ( b ) BC is thought of as being made of a pile of(infinitely thin) rings parallel to the opposing substrate andwith an area d S ( r ) = pr d r , where r is the radius of thering. Each of these rings is partly facing (in normal direc-tion) the surface with ( a < ) BC, with an extension d S < ( r ) , andpartly facing the surface with ( a > ) BC on the other side of thechemical step [Fig. 14], with an extension d S > ( r ) , such thatd S ( r ) = d S < ( r ) + d S > ( r ) . For an assigned r , d S ≷ ( r ) de-pend, inter alia, on the lateral position X of the colloid. Usingthe assumption of additivity of the forces underlying the DAwe suppose that the contribution d F s ( r ) of the ring to the to-tal critical Casimir force F s is given by the sum of the criticalCasimir forces which would act, in a film, on portions of ar-eas d S < and d S > in the presence of ( a < , b ) and ( a > , b ) BC,respectively. According to Eq. (1) this leads to the followingexpression for the force acting on a single ring:d F s ( r ) k B T = d S < ( r ) L d ( r ) k ( a < , b ) ( sign ( t ) L ( r ) / x ± )+ d S > ( r ) L d ( r ) k ( a > , b ) ( sign ( t ) L ( r ) / x ± ) , (A1)where L ( r ) is the substrate-ring distance [Fig. 14] as givenin Eq. (7), and k ( a ≷ , b ) are the scaling functions of the criticalCasimir force in the film geometry with ( a > , b ) and ( a < , b ) BC, respectively [see Eq. (1)]. This assumption neglects alledge effects along the boundary between the areas d S > ( r ) andd S < ( r ) , which might actually be relevant in view of the spa-tial variation of the order parameter profile. It is therefore im-portant to test the validity of this assumption at least in somerelevant cases. This is carried out in Sec. IV for d =
4, i.e.,within MFT.Without loss of generality in the following we assume X >
0, i.e., that the normal projection of the center of the spherefalls on the part of the substrate with ( a > ) BC [Figs. 1 and14]. The results for X < a < ↔ a > and X ↔ − X . Taking into accountthat d S ( r ) = d S < ( r ) + d S > ( r ) one can rewrite Eq. (A1) asd F s ( r ) k B T = d S ( r ) L d ( r ) k ( a > , b ) ( sign ( t ) L ( r ) / x ± )+ d S < ( r ) L d ( r ) D k ( sign ( t ) L ( r ) / x ± ) , (A2)where D k ( Q ) = k ( a < , b ) ( Q ) − k ( a > , b ) ( Q ) . Summing up all forcecontributions from the rings of different radii r , one finds for (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) L ( ρ ) D ( a > )( a < ) R d S ( ρ )d S > ( ρ )d S < ( ρ ) ρX FIG. 14. Sketch concerning the Derjaguin approximation for the crit-ical Casimir force acting on a colloid opposite to a chemical step.The critical Casimir force is subdivided into contributions from ringsparallel to the substrate. The projection of the area d S ( r ) of a ringonto the substrate is separated into the areal contributions d S < andd S > which emerge as the intersection of the projected ring with thehalf-planes carrying ( a < ) and ( a > ) BC, respectively [see the maintext]. The sphere has a surface-to-surface distance D from the sub-strate and its center has a lateral distance X from the chemical step. the total normal force acting on the sphere F s ( X , D , R , T ) = F ( a > , b ) ( D , R , T ) + D F ( X , D , R , T ) , (A3)where F ( a > , b ) is the force acting on a sphere close to a homo-geneous substrate with ( a > ) BC and is given by Eq. (9) or byEqs. (5) and (10). This term does not contribute to the lat-eral critical Casimir force experienced by the colloid near thechemical step, because it does not depend on the lateral co-ordinate of the colloid. The second term D F in Eq. (A3) cor-responds to the integration of the force differences D k in theregion of overlap between the projection of the sphere ontothe substrate plane and that part of the substrate with ( a < ) BC. For each ring this area is given by [see Fig. 14]d S < ( r ) = ( , r < X , ( X / r ) r d r , X ≤ r ≤ R . (A4)This leads to D F ( X , D , R , T ) k B T = R Z X d rr arccos (cid:18) X r (cid:19) D k ( sign ( t ) L ( r ) / x ± ) L d ( r ) . (A5)In the spirit of the DA, the radius of the sphere is taken tobe large compared to its distance to the substrate, i.e., D = D / R ≪
1, and the contributions from the rings closest to thesubstrate dominate. Therefore, it is well justified and in accor-dance with the DA to assume X / R ≪ parabolic approximation for the distance of the ringsto the substrate [Eq. (7)], L ( r ) ≃ D a , with a = + r / RD .Changing the integration variable in Eq. (A5) we directly find D F ( X , D , R , T ) = k B T RD d − D K ( X , Q , D ) , (A6)2where D K is a universal scaling function given by D K ( X , Q , D → ) = ¥ Z + X / d aa − d arccos (cid:18) X √ ( a − ) (cid:19) D k ( a Q ) . (A7)Note that the relevant scaling variable X = X / √ RD cantake on arbitrary values, irrespective of the two assumptions D / R ≪ X / R ≪
1. From Eq. (A7) one finds withEqs. (10) and (13) directly the expression for the scaling func-tion y ( a < | a > , b ) given in Eq. (16).The critical Casimir potential F s ( X , D , R , T ) = R ¥ D d lF s ( X , l , R , T ) can be separated analogously to Eq. (A3),i.e., F s ( X , D , R , T ) = F ( a > , b ) ( D , R , T ) + DF ( X , R , D , T ) (A8)with DF ( X , R , D , T ) = ¥ Z D d l D F ( X , l , R , T ) = : k B T RD d − D J ( X , Q , D ) . (A9)Using Eq. (A7), the scaling function D J is given by
D J ( X , Q , D ) = ¥ Z d y y d − ¥ Z + X / ( y ) d a a d arccos X p y ( a − ) ! D k ( y a Q ) . (A10)By changing the integration variable a z : = y ( a − ) / X followed by y v : = y + X z / D J ( X , Q , D ) = X Z ¥ d z Z ¥ + z X / d v v d arccos ( / √ z ) D k ( v Q ) . (A11)After changing the order of integration Z ¥ d z Z ¥ + z X / d v = Z ¥ + X / d v Z ( v − ) / X d z , (A12)and using the primitive Z d z arccos ( / √ z ) = z arccos ( / √ z ) − √ z − + c , (A13)one obtains after a final change of variables v w : = ( v − ) / X D J ( X , Q , D ) = X Z ¥ d s ( + X s / ) d h s arccos ( s − / ) − √ s − i D k ( Q [ + X s / ]) . (A14)From Eq. (A14) together with Eq. (11) one obtains the final expression for the scaling function of the critical Casimir potentialas given in Eq. (18).
1. Bulk critical point: Q = In order to calculate the critical Casimir force acting on the colloid at the bulk critical point one inserts Eq. (2) into Eq. (A7)and obtains D K ( X , Q = , D ) = (cid:0) D ( a < , b ) − D ( a > , b ) (cid:1) Z ¥ + X / d a a − d arccos (cid:18) X √ a − (cid:19) (A15) = : X (cid:0) D ( a < , b ) − D ( a > , b ) (cid:1) I d ( X / ) , where D ( a , b ) = k ( a , b ) ( ) [see Eq. (2)], and with the substitution a z = X / p ( a − ) for d > I d ( a ) = Z d z z d − ( z + a ) d arccos ( z ) . (A16) For I d ( a ) the recursion relation I d + ( a ) = d a − d dd a [ a d I d ( a )] (A17)holds, so that I and I can be expressed in terms of I . Per-3forming the integration we find I ( a ) = p a " − a / ( + a ) / , (A18)and therefore with Eq. (A17) I ( a ) = p a " − a / + a / ( + a ) / , (A19)and I ( a ) = p a " − a / + a / + a / ( + a ) / . (A20)Thus, from Eqs. (A16), (A19), and (A20) together with theexpression for K ( a ≷ , b ) ( , ) = p D ( a ≷ , b ) / ( d − ) [Sec. III A]and Eq. (13), one finds the expression for the scaling function y ( a < | a > , b ) given in Eq. (17). The critical Casimir potential at Q = d = D J ( X , Q = , D ) = X (cid:0) D ( a < , b ) − D ( a > , b ) (cid:1) Z ¥ d y y − d I d (cid:16) X y (cid:17) , (A21)and from a change of variable y a = X / ( y ) one finds D J ( X , , D ) = d − X d − (cid:0) D ( a < , b ) − D ( a > , b ) (cid:1) Z X / d a a d − I d ( a ) . (A22)Using Eq. (A17) and the limiting behavior I d ( a → ) = p / ( ( d − ) a ) , we find D J ( X , , D ) = X ( d − ) (cid:0) D ( a < , b ) − D ( a > , b ) (cid:1) I d − ( X / ) . (A23)From Eqs. (A18), (A19), and (A23) together with J ( , ) asgiven in Sec. III A one obtains Eq. (19) for the scaling functionof the critical Casimir potential at T c .
2. Far from criticality: Q ≫ Far from the critical point, i.e., for Q ≫
1, and for symmetry breaking boundary conditions ( a < ) = (+) , ( a > ) = ( − ) , and ( b ) = ( − ) Eq. (3) holds and the integrals in Eqs. (A7) and (A14) can be calculated analytically. For Q ≫ D K ( X , Q ≫ , D ) = ( A − − A + ) Q d Z ¥ + X / d a arccos (cid:18) X √ ( a − ) (cid:19) e − a Q . (A24)Substituting a b = ( a − ) / X one has D K ( X , Q ≫ , D ) = X ( A − − A + ) Q d e − Q Z ¥ d b arccos ( b − ) e − X Q b / . (A25)Integrating by parts leads to D K ( X , Q ≫ , D ) = ( A − − A + ) Q d − e − Q Z ¥ d b b p b − e − X Q b / . (A26)By using the relation Z ¥ d b b p b − e − a b = p erfc ( a ) , (A27)where a > ( a ) = − erf ( a ) = p − / R ¥ a d t exp ( − t ) is the complementary error function, we finally arrive at D K ( X , Q ≫ , D ) = p ( A − − A + ) Q d − e − Q erfc ( X p Q / ) . (A28)The scaling function K ( ∓ , − ) for Q ≫ K ( ∓ , − ) ( Q ≫ , D → ) = p A ± Q d − e − Q (A29)and from Eqs. (13), (A28), and (A29) one obtains the expression for y ( −| + , − ) as given in Eq. (20). Similarly, after rewritingEq. (A14) for Q ≫ D J ( X , Q ≫ , D ) = ( A − − A + ) Q d e − Q X Z ¥ d s (cid:16) s arccos ( s − / ) − √ s − (cid:17) e − X Q s / , (A30)one can integrate by parts, which yields D J ( X , Q ≫ , D ) = ( A − − A + ) Q d − e − Q Z ¥ d s √ s − (cid:20) s + QX (cid:0) + QX (cid:1) − Q X s (cid:21) e − X Q s / . (A31)4Using Eq. (A27) and the relations [which follow from taking successive derivatives − d / d ( a ) of Eq. (A27)] Z ¥ d s √ s − e − a s = √ p a e − a , Z ¥ d s s √ s − e − a s = √ p a (cid:0) + a (cid:1) e − a , (A32)one ends up with D J ( X , Q ≫ , D ) = p ( A − − A + ) Q d − e − Q erfc ( X p Q / ) . (A33)Together with the expression for the homogeneous case [see Sec. III and Ref. 7], J ( ∓ , − ) ( Q ≫ , D → ) = p A ± Q d − e − Q , (A34)one obtains the expression for w ( −| + , − ) given in Eq. (20). Appendix B: Derjaguin approximation for a single chemical lane
Based on the assumption of additivity which underlies the Derjaguin approximation one can use the results presented in Sec. IVfor a chemical step in order to study a chemical lane. The chemical lane configuration can be regarded as the superposition oftwo chemical steps, one ( A ) being a chemical step located at x = − L with ( a | a ℓ ) BC, and the other one ( B ) being a chemicalstep located at x = L with ( a ℓ | a ) BC. This superposition overcounts a contribution corresponding to a homogeneous substratewith ( a ℓ ) BC which must be subtracted [see Eq. (13)]: ( A ) : ( a ) | − L ( a ℓ ) :::::::: +( B ) : ( a ℓ ) :::::::: | L ( a ) − :::: ( a ℓ ) :::::::: = ( a ) | − L ( a ℓ ) :::: | L ( a ) K ( A ) s + K ( B ) s − K ( a ℓ , b ) = K ℓ , (B1)where K ( A ) s ( L , X , Q , D ) = K ( a , b ) + K ( a ℓ , b ) + K ( a , b ) − K ( a ℓ , b ) y ( a | a ℓ , b ) ( X + L , Q , D ) (B2)and K ( B ) s ( L , X , Q , D ) = K ( a , b ) + K ( a ℓ , b ) + K ( a ℓ , b ) − K ( a , b ) y ( a ℓ | a , b ) ( X − L , Q , D ) . (B3)Since within the DA y ( a ℓ | a , b ) = y ( a | a ℓ , b ) , Eqs. (B1)–(B3) and Eq. (27) lead directly to Eq. (29). The procedure for calculatingthe critical Casimir potential is analogous to the one discussed here for the force and leads to Eq. (30). Appendix C: Derjaguin approximation for periodic chemical patterns
In order to obtain the scaling function for the critical Casimir force and the potential of a sphere close to a periodic chemicalpattern one can follow a procedure analogous to the one presented in Appendix B. Indeed, in order to form a lane ℓ ′ with ( a ) BC on an otherwise homogeneous portion of a substrate with ( a ) BC and lateral extension P , one can proceed as follows:(A): superimpose onto the substrate the single chemical lane ℓ studied in Sec. V, with a ℓ = a , a = a , suitably positioned inspace such that it coincides with the lane ℓ ′ to be formed.(B): subtract the contribution of a homogeneous substrate with ( a ) BC, which is overcounted in the previous superposition.After this subtraction, the contribution to the force resulting from that part – marked by ( ? ) in Eq. (C1) – of the originalsubstrate which is not affected by the formation of the extra lane is unchanged.5 ( ? ) QPPPPPPR P z }| { ( a ) ( ? ) QPPPPPPR ( A ) : + ( a ) | X ′ − L ( a ) :::: | X ′ + L ( a ) ( B ) : − ( a ) = ( ? ) QPPPPPPR ( a ) | X ′ − L ( a ) :::: | X ′ + L ( a ) ( ? ) QPPPPPPR lane ℓ ′ (C1)The contribution D F to the critical Casimir force experienced by a colloid close to such a substrate and due to the addition ofthe lane is characterized by the scaling function [see Eq. (27)] D K ( l , P , X − X ′ , Q , D → ) = K ℓ ( P l , X − X ′ , Q ) − K ( a , b ) = K ( a , b ) − K ( a , b ) × h y ( a | a , b ) ( X − X ′ + P l , Q , D → ) − y ( a | a , b ) ( X − X ′ − P l , Q , D → ) i (C2)where we have used the relation ( L / ) / √ RD = P l / X ′ ≡ X ′ / √ RD , with X ′ as the position of the centerof the added lane ℓ ′ . The force resulting from a periodic pattern can now be obtained by starting out with a homogeneoussubstrate with ( a ) BC and by iterating the procedure discussed above which adds progressively displaced lanes at positions X ′ = nP , i.e., X ′ = n P , with n ∈ Z . The resulting force is characterized by the scaling function K p ( l , P , X , Q , D → ) = K ( a , b ) + + ¥ (cid:229) n = − ¥ D K ( l , P , X − n P , Q , D → ) (C3)which, together with Eq. (33), yields immediately Eq. (35) for y p .For l = l = ( a , b ) BC or ( a , b ) BC, respectively. Obviously,for l =
0, the sum in Eq. (35) vanishes, and one is left with y p ( l = , P , X , Q , D → ) =
1, corresponding to ( a , b ) BC. On theother hand for l =
1, the sum in Eq. (35) can be easily evaluated [see Eq. (16) for | X | → ¥ ]:lim M , N → ¥ N (cid:229) n = − M (cid:8) y ( a | a , b ) ( X + P ( n + ) , Q , D ) − y ( a | a , b ) ( X + P ( n − ) , Q , D ) (cid:9) = lim M , N → ¥ (cid:8) y ( a | a , b ) ( X + P ( N + ) , Q , D ) − y ( a | a , b ) ( X + P ( − M − ) , Q , D ) (cid:9) = − , (C4)where we have used the fact that y ( a | a , b ) ( X = ± ¥ , Q , D ) = ∓
1. Accordingly, y p ( l = , P , X , Q , D → ) = −
1, which corre-sponds to the homogeneous case with ( a , b ) BC.In the limit P → √ RD ), the sum in Eq. (35) turns into an integral: ¥ (cid:229) n = − ¥ n y ( a | a , b ) ( X + P ( n + l ) , Q , D ) − y ( a | a , b ) ( X + P ( n − l ) , Q , D ) o −−−→ P → P Z ¥ − ¥ d h n y ( a | a , b ) ( X + h + P l , Q , D ) − y ( a | a , b ) ( X + h − P l , Q , D ) o = Z ¥ − ¥ d h l dd h y ( a | a , b ) ( X + h , Q , D )= l (cid:8) y ( a | a , b ) (+ ¥ , Q , D ) − y ( a | a , b ) ( − ¥ , Q , D ) (cid:9) = − l , (C5)and finally one finds Eq. (36).For completeness, we provide the corresponding expression for the scaling function of the critical Casimir potential w p withinthe DA: w p ( l , P , X , Q , D → ) = + ¥ (cid:229) n = − ¥ n w ( a | a , b ) ( X + P ( n + l ) , Q , D → ) − w ( a | a , b ) ( X + P ( n − l ) , Q , D → ) o . (C6)In the limit P → w p reduces to w p ( l , P → , X , Q , D → ) = − l . (C7)6Accordingly, within the DA and in the limit P → J p ( l , P → , X , Q , D → ) = lJ ( a , b ) ( Q , D → ) + ( − l ) J ( a , b ) ( Q , D → ) . (C8) Appendix D: Cylinder close to a patterned substrate1. Derjaguin approximation for a homogeneous substrate
Similarly to the case of a sphere discussed before, the crit-ical Casimir force F cyl ( a , b ) per unit length acting on a (three-dimensional) cylinder of radius R with ( b ) BC close to andparallel to a substrate with ( a ) BC at a surface-to-surface dis-tance D can be expressed in terms of a universal scaling func-tion K cyl : F cyl ( a , b ) ( D , R , T ) = k B T R / D d − / K cyl ( a , b ) ( Q , D ) , (D1)with Q = sign ( t ) D / x ± and D = D / R as before. Equation (D1)describes a force divided by a length and per D d − which for d = F cyl ( a , b ) per length of its axisand per length of the extra translationally invariant directionof a hypercylinder [compare Eq. (5)]. The geometric prefac-tor in Eq. (D1), however, differs from the one for the sphere[Eq. (5)] because it is chosen such that within the DA ( D → K cyl ( a , b ) attains a nonzero and finite limit, asdiscussed before. The DA can be implemented along the linesof Sec. III A for the sphere. Here the surface of the cylindricalcolloid is decomposed into pairs of infinitely narrow stripes ofcombined area d S = M d r , positioned parallel to the substrateat a distance L ( r ) from it [Eq. (7)] and each at a distance r from the symmetry plane of the configuration. M is the lengthof the cylinder and drops out from F cyl ( a , b ) which follows analo-gously from Eqs. (8) and (9): F cyl ( a , b ) ( D , R , T ) / k B T ≃ Z R d r [ L ( r )] − d k ( a , b ) ( sign ( t ) L ( r ) / x ± ) , (D2)where L ( r ) is given in Eq. (7). Finally, in the limit D → K cyl ( a , b ) ( Q , D → ) = √ ¥ Z d a ( a − ) − a − d k ( a , b ) ( Q a ) . (D3)At the bulk critical point Q = K cyl ( a , b ) ( , ) = √ p [ G ( d − ) / G ( d )] D ( a , b ) so that K cyl ( a , b ) ( , ) = [ p / ( √ )] D ( a , b ) ≃ . × D ( a , b ) for d = K cyl ( a , b ) ( , ) = [ p / ( √ )] D ( a , b ) ≃ . × D ( a , b ) for d =
2. Derjaguin approximation for a chemical step
Here, we assume that the axis of the cylinder is parallel tothe chemical step, i.e., perpendicular to the x direction [Fig. 1],as well as parallel to the substrate. The projection of the posi-tion of the axis of the cylinder with respect to the x direction is denoted by X , so that at X = F cyls ( X , D , R , T ) = F cyl ( a > , b ) ( D , R , T )+ D F cyl ( X , D , R , T ) . (D4)Within the DA we find for D → D F cyl ( X , D , R , T ) = k B T R / D d − / D K cyl ( X , Q , D → ) , (D5)where [compare Eq. (A7)] D K cyl ( X , Q , D → ) = √ Z ¥ + X / d a ( a − ) − a − d D k ( Q a ) . (D6)Using Eq. (D6) and Eq. (D3) we find for the whole range ofvalues of X the scaling function y cyl ( a < | a > , b ) which is definedcompletely analogous to Eq. (13) [compare Eq. (16)]: y cyl ( a < | a > , b ) ( X ≷ , Q , D → ) = ∓ ± √ R ¥ + X / d a ( a − ) − a − d D k ( Q a ) K cyl ( a < , b ) ( Q , D → ) − K cyl ( a > , b ) ( Q , D → ) . (D7)
3. Derjaguin approximation for a periodic chemical pattern
The derivation of the scaling function for the criticalCasimir force acting on the cylinder close to and aligned witha periodic chemical pattern as studied in Sec. VII is analogousto the one for the sphere described in Appendix C. The finalformula for y cylp is the same as in Eq. (35) with y ( a | a , b ) re-placed by y cyl ( a | a , b ) given by Eq. (D7). This renders the criticalCasimir force per unit length F cylp ( L , P , X , D , R , T ) = k B T R / D d − / K cylp ( l , P , X , Q , D ) (D8)where K cylp is defined as in Eq. (33) with K ( a , b ) and K ( a , b ) replaced by K cyl ( a , b ) and K cyl ( a , b ) , respectively, which are givenby Eq. (D3), and with y p replaced by y cylp . The correspondingresults are shown in Fig. 12.7 H. G. B. Casimir, Proc. K. Ned. Akad. Wet. , 793 (1948). M. Kardar and R. Golestanian, Rev. Mod. Phys. , 1233 (1999). M. E. Fisher and P. G. de Gennes, C. R. Acad. Sci., Paris, Ser. B , 207(1978). A. Gambassi, J. Phys.: Conf. Ser. , 012037 (2009). A. Gambassi, C. Hertlein, L. Helden, C. Bechinger, and S. Dietrich, Euro-physics News , 18 (2009). C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, Nature , 172 (2008). A. Gambassi, A. Maciołek, C. Hertlein, U. Nellen, L. Helden, C. Bechinger,and S. Dietrich, Phys. Rev. E , 061143 (2009). M. Krech,
The Casimir Effect in Critical Systems (World Scientific, Singa-pore, 1994). J. G. Brankov, D. M. Danchev, and N. S. Tonchev,
Theory of critical phe-nomena in finite size systems (World Scientific, Singapore, 2000). M. Krech and S. Dietrich, Phys. Rev. Lett. , 345 (1991); Phys. Rev. A ,1886 (1992); ibid 1922 (1992). R. Evans and J. Stecki, Phys. Rev. B , 8842 (1994). H. W. Diehl, D. Grüneberg, and M. A. Shpot, EPL , 241 (2006). R. Zandi, A. Shackell, J. Rudnick, M. Kardar, and L. P. Chayes, Phys. Rev.E , 030601 (2007). F. M. Schmidt and H. W. Diehl, Phys. Rev. Lett. , 100601 (2008). T. F. Mohry, A. Maciołek, and S. Dietrich, arXiv:1004.0112 (2010);T. F. Mohry, diploma thesis, University of Stuttgart (2008). J. N. Munday, F. Capasso, and V. A. Parsegian, Nature , 170 (2009). R. Garcia and M. H. W. Chan, Phys. Rev. Lett. , 1187 (1999); ibid ,086101 (2002). A. Ganshin, S. Scheidemantel, R. Garcia, and M. H. W. Chan, Phys. Rev.Lett. , 075301 (2006). M. Fukuto, Y. F. Yano, and P. S. Pershan, Phys. Rev. Lett. , 135702(2005). S. Rafai, D. Bonn, and J. Meunier, Physica A , 31 (2007). A. Hucht, Phys. Rev. Lett. , 185301 (2007). O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, EPL , 60009(2007). O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, Phys. Rev. E ,041142 (2009); ibid , 039902(E) (2009). M. Hasenbusch, J. Stat. Mech., P07031 (2009); Phys. Rev. E , 061120(2009); arXiv:0907.2847 (2009). T. W. Burkhardt and E. Eisenriegler, Phys. Rev. Lett. , 3189 (1995); ibid , 2867 (1997). E. Eisenriegler and U. Ritschel, Phys. Rev. B , 13717 (1995). A. Hanke, F. Schlesener, E. Eisenriegler, and S. Dietrich, Phys. Rev. Lett. , 1885 (1998). F. Schlesener, A. Hanke, and S. Dietrich, J. Stat. Phys. , 981 (2003). E. Eisenriegler, J. Chem. Phys. , 3299 (2004). S. Kondrat, L. Harnau, and S. Dietrich, J. Chem. Phys. , 204902 (2009). M. Tröndle, S. Kondrat, A. Gambassi, L. Harnau, and S. Dietrich, EPL ,40004 (2009). F. Soyka, O. Zvyagolskaya, C. Hertlein, L. Helden, and C. Bechinger, Phys.Rev. Lett. , 208301 (2008). M. Sprenger, F. Schlesener, and S. Dietrich, J. Chem. Phys. , 134703(2006). M. Tröndle, L. Harnau, and S. Dietrich, J. Chem. Phys. , 124716 (2008). M. Krech, Phys. Rev. E , 1642 (1997). U. Leonhardt and T. G. Philbin, New J. Phys. , 254 (2007). A. W. Rodriguez, J. N. Munday, J. D. Joannopoulos, F. Capasso, D. A. R.Dalvit, and S. G. Johnson, Phys. Rev. Lett. , 190404 (2008). A. W. Rodriguez, A. P. McCauley, D. Woolf, F. Capasso, J. D. Joannopou-los, and S. G. Johnson, Phys. Rev. Lett. , 160402 (2010). S. J. Rahi, M. Kardar, and T. Emig, arXiv:0911.5364 (2009). S. J. Rahi and S. Zaheer, arXiv:0909.4510 (2009). R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, Phys.Rev. Lett. , 103602 (2009). K. Binder, in
Phase Transitions and Critical Phenomena , edited byC. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 8, p. 1. H. W. Diehl, in
Phase Transitions and Critical Phenomena , edited byC. Domb and J. L. Lebowitz (Academic, London, 1986), Vol. 10, p. 75. H. W. Diehl, Int. J. Mod. Phys. B , 3503 (1997). A. Pelissetto and E. Vicari, Phys. Rep. , 549 (2002). V. Privman, P. C. Hohenberg, and A. Aharony, in
Phase Transitions andCritical Phenomena , edited by C. Domb and J. L. Lebowitz (Academic,London, 1991), Vol. 14, p. 1 and p. 364. H. B. Tarko and M. E. Fisher, Phys. Rev. Lett. , 926 (1973); Phys. Rev.B , 1217 (1975). Z. Borjan and P. J. Upton, Phys. Rev. Lett. , 125702 (2008). T. W. Burkhardt and H. W. Diehl, Phys. Rev. B , 3894 (1994). H. W. Diehl and M. Smock, Phys. Rev. B , 5841 (1993); ibid , 6740(1993). B. Derjaguin, Kolloid Z. , 155 (1934). For the scaling function in d = ( ± ) BC, we use the approximation denoted by ( i ) in Figs. 9 and 10 of Ref. 23. The uncertainty of the overall amplitudeof the scaling functions is about 10% to 20% as indicated by the differ-ent results obtained by the various approximations used in Ref. 23. Corre-spondingly, this uncertainty affects our predictions for the scaling functions K ( ± , − ) , J ( ± , − ) , K k s , K p , F p , and K cylp based on such Monte Carlo simulationdata. However, the normalized scaling functions w (+ |− , − ) , w ℓ , y p , and y cylp are affected less leading to an uncertainty of at most 3%. U. Nellen, L. Helden, and C. Bechinger, EPL , 26001 (2009). D. Vogt, diploma thesis, University of Stuttgart (2009). Private communication by D. Vogt, O. Zvyagolskaya, and C. Bechinger. A. W. Rodriguez, D. Woolf, A. P. McCauley, F. Capasso, J. D. Joannopou-los, and S. G. Johnson, arXiv:1004.2733 (2010). M. Hoffmann, Y. Lu, M. Schrinner, M. Ballauff, and L. Harnau, J. Phys.Chem. B , 14843 (2008); M. Hoffmann, M. Siebenbürger, L. Harnau,M. Hund, C. Hanske, Y. Lu, C. S. Wagner, M. Drechsler, and M. Ballauff,Soft Matter , 1125 (2010). M. Sprenger, F. Schlesener, and S. Dietrich, Phys. Rev. E , 056125(2005). D. Dantchev, H. W. Diehl, and D. Gruneberg, Phys. Rev. E , 016131(2006). D. Dantchev, F. Schlesener, and S. Dietrich, Phys. Rev. E , 011121(2007). A. Maciołek, A. Gambassi, and S. Dietrich, Phys. Rev. E , 031124(2007). See Eq. 7 . . Tables of indefinite integrals , edited by Y. A.Brychkov, O. I. Marichev, and A. P. Prudnikov (Gordon and Breach, NewYork, 1989), with the substitution z x = / √ z . See Eq. (4) of Tab. 234 in
Nouvelles tables d’intégrales définies , editedby D. B. De Haan (P. Engels, Leide, 1867); note that there is a mis-print in Eq. 4 . . Table of Integrals, Series, and Products , Sixthedition, edited by I. S. Gradshteyn and I. M. Ryzhik (Academic, Lon-don, 2000). The correct expression is R d x x (cid:0) arccos ( x ) (cid:1) / ( + qx ) = p ( √ + q − ) / ( q √ + q ) for q > − See Eq. (26) on p. 136 in