document: the first step in describing the interaction between many particles is to determine their pair potential or the forces among a single pair . 
   if the governing physical equations are linear ( like for gravity or electrostatics ) , this approach yields a quantitatively reliable description of the physical system considered , based on the linear superposition principle . 
   however , if nonlinearities are present , linear superposition of pair potentials is no longer accurate and nonadditivity gives rise to many - body effects . 
   these latter effects can lead , e.g. , to a strengthening or weakening of the total force acting on a particle surrounded by more than a single other one , a change of sign of that force , or the appearance of stable or unstable configurations . 
   many - body effects appear in rather diverse systems such as nuclear matter  , superconductivity  @xcite , colloidal suspensions  @xcite , quantum - electrodynamic casimir forces  @xcite , polymers  @xcite , nematic colloids  @xcite , and noble gases with van der waals forces acting among them  @xcite . each of these systems is characterized by a wide range of time and length scales . integrating out the degrees of freedom associated with small scales ( such as the solvent of colloidal solutes or polymers ) for fixed configurations of the large particles , generates effective interactions among the latter , which are inherently not pairwise additive . 
   this is the price to be paid for achieving a reduced description of a multicomponent system . driven by these effective interactions the large particles of the system may exhibit collective behavior of their own ( like aggregation or phase separation , see refs . 
   @xcite and  @xcite and references therein ) , which can be described much easier if it is governed by pair potentials . in order to be able to judge whether this ansatz is adequate 
   one has to check the relative magnitude of genuine many - body forces .    in this paper 
   we assess the quantitative influence of such many - body effects on critical casimir forces ( ccfs )  @xcite . 
   these long - ranged forces arise as a consequence of the confinement of the order parameter fluctuations in a critical fluid  @xcite . 
   they have been analyzed paradigmatically by studying the effective interaction between a _ 
   single _ colloidal particle and a homogeneous  @xcite or inhomogeneous  @xcite container wall as well as between _ two isolated _ colloidal particles  @xcite upon approaching the critical point of the solvent . here 
   we add one sphere to the sphere - wall configuration , which is the simplest possibility to study many - body forces . 
   ( the wall mimics a third , very large sphere . )    in order to be able to identify the latter ones one has to resort to a theoretical scheme which allows one to compute the forces between individual pairs and the three - body forces on the same footing . 
   since these forces are characterized by universal scaling functions , which depend on the various geometrical features of the configuration and on the thermodynamic state , we tackle this task by resorting to field - theoretic mean field theory ( mft ) , which captures the universal scaling functions as the leading contribution to their systematic expansion in terms of @xmath1 spatial dimensions . 
   experience with corresponding previous studies for simple geometries tells that this approach does yield the relevant qualitative features of the actual universal scaling functions in @xmath2 ; if suitably enhanced by renormalization group arguments these results reach a semi - quantitative status . 
   we point out that even within this approximation the numerical implementation of this corresponding scheme poses a severe technical challenge . 
   thus at present this approach appears to be the only feasible one to explore the role of many - body critical casimir forces within the full range of their scaling variables . 
   accordingly we consider the standard landau - ginzburg - wilson hamiltonian for critical phenomena of the ising bulk universality class , which is given by    @xmath3 = \int_v{\rm d}^d\mathbf{r } \left\ { \frac{1}{2}\left ( \nabla\phi \right)^2 + \frac{\tau}{2}\phi^2 + \frac{u}{4!}\phi^4 \right\ } ~,\ ] ]    with suitable boundary conditions ( bcs ) . in the case of a binary liquid mixture near its consolute ( demixing ) point , the order parameter @xmath4 is proportional to the deviation of the local concentration of one of the two species from the critical concentration . 
   @xmath5 is the volume accessible to the fluid , @xmath6 is proportional to the reduced temperature @xmath7 , and the coupling constant @xmath8 stabilizes the statistical weight @xmath9 in the two - phase region , i.e. , for @xmath10 . 
   close to the bulk critical point @xmath11 the bulk correlation length @xmath12 diverges as @xmath13 , where @xmath14 in @xmath2 and @xmath15 in @xmath16 , i.e. , within mft  @xcite . 
   the two non - universal amplitudes @xmath17 are of molecular size ; they form the universal ratio @xmath18 for @xmath2 and @xmath19 for @xmath16  @xcite . 
   the bcs reflect the generic adsorption preference of the confining surfaces for one of the two components of the mixture . for the critical adsorption fixed point  @xcite , 
   the bc at each of the confining surfaces is either @xmath20 or @xmath21 , to which we refer as @xmath22 or @xmath23 , respectively .    within mft 
   the equilibrium order parameter distribution minimizes the hamiltonian in eq .   for the aforementioned bcs , 
   i.e. , @xmath24/\delta\phi = 0 $ ] . 
   far from any boundary the order parameter approaches its constant bulk value @xmath25 for @xmath10 or @xmath26 for @xmath27 . 
   @xmath28 is a non - universal bulk amplitude and @xmath29 ( for @xmath16 ) is a standard critical exponent . in the following we consider the reduced order parameter @xmath30 . 
   the remainder of this paper is organized as follows . in sec . 
   [ section_the_system ] we define the system under consideration and the scaling functions for the ccfs as well as the normalization scheme . in sec . 
   [ section_results ] we present the numerical results obtained for the universal scaling functions of the ccfs , from which we extract and analyze the many - body effects . in sec . 
   [ section_conclusion ] we summarize our results and draw some conclusions . 
   we study the normal and the lateral ccfs acting on two colloidal particles immersed in a near - critical binary liquid mixture and close to a homogeneous , planar substrate . 
   we focus on the critical concentration which implies the absence of a bulk field conjugate to the order parameter [ see eq .  ] . 
   the surfaces of the colloids and of the substrate are considered to exhibit a strong adsorption preference for one of the two components of the confined liquid leading to @xmath31 bcs . 
   the forces are calculated using the full three - dimensional numerical analysis of the appropriate mft as given by eq .  . 
   specifically , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a homogeneous substrate with bc @xmath36 at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively ( see fig .  [ system_sketch ] ) . 
   the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath40 and @xmath41 so that the distance between the centers , projected onto the @xmath42-axis , is given by @xmath43 . 
   the bcs of the whole system are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . 
   it is important to point out that we discuss colloidal particles with the shape of a hypercylinder    @xmath49    where @xmath50 are the semiaxes ( or radii ) of the hypercylinder and @xmath51 , @xmath52 , @xmath53 . 
   if @xmath54 and @xmath55 , the hypercylinder reduces to a hypersphere . 
   the generalization of @xmath56 to values larger than 3 is introduced for technical reasons because @xmath57 is the upper critical dimension for the relevance of the fluctuations of the order parameter . 
   these fluctuations lead to a behavior different from that obtained from the present mft which ( apart from logarithmic corrections  @xcite ) is valid in @xmath16 . 
   we consider two hypercylinders in @xmath16 with @xmath58 and @xmath59 . 
   the two colloids are taken to be parallel along the fourth dimension with infinitely long hyperaxes in this direction . 
   considering hypercylinders , which are translationally invariant along the @xmath60axis , allows us to minimize @xmath61 $ ] numerically using a three dimensional finite element method in order to obtain the spatially inhomogeneous order parameter profile @xmath62 for the geometries under consideration ( see fig . 
   [ system_sketch ] ) . 
   and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . 
   the two colloidal particles with bcs @xmath34 and @xmath64 are located at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively . 
   the substrate exhibits bc @xmath36 . 
   the lateral distance between the centers of the spheres along the @xmath42-direction is given by @xmath43 , while the centers of both spheres lie in the plane @xmath65 . 
   in the case of four spatial dimensions the figure shows a cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 . ]    in the case of an upper critical demixing point of the binary liquid mixture at the critical concentration , @xmath67 corresponds to the disordered ( i.e. , mixed ) phase of the fluid , whereas @xmath68 corresponds to the ordered ( i.e. , phase separated ) phase . 
   the meaning of the sign is reversed for a lower critical point . in the following we assume an upper critical point . 
   the normal ccf @xmath69 acting on sphere @xmath70 in the presence of sphere @xmath71 ( @xmath72 and @xmath73 ) along the @xmath74-direction takes the scaling form    @xmath75    where @xmath76 , @xmath77 , @xmath78 , @xmath79 , and @xmath80 ( i.e. , @xmath81 for @xmath27 and @xmath82 for @xmath10 ) . 
   equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . 
   @xmath83 is the force per length of the hypercylinder due to its extension in the translationally invariant direction . in the spirit of a systematic expansion in terms of @xmath84 around the upper critical dimension we study the scaling functions @xmath85 within mft as given by eq .   for hypercylinders in @xmath16 , which captures the correct scaling functions in @xmath16 up to logarithmic corrections occurring in @xmath86  @xcite , which we do not take into account here . 
   since mft renders the leading contribution to an expansion around @xmath16 , geometrical configurations with small @xmath87 , @xmath88 , or @xmath89 are not expected to be described reliably by the present approach due to the dimensional crossover in narrow slit - like regions , which is not captured by the @xmath84 expansion . 
   the colloidal particles will also experience a lateral ccf @xmath90 , for which it is convenient to use the scaling form    @xmath91    where @xmath92 ( i.e. , @xmath93 for @xmath27 and @xmath94 for @xmath10 ) . 
   note that the choice of @xmath95 as the scaling variable does not depend on the type of particle the force acts on . 
   equation also describes the singular contribution to the lateral force near @xmath11 . 
   the total ccf acting on particle @xmath70 is    @xmath96    where @xmath97 and @xmath98 are the unit vectors pointing in @xmath42- and @xmath74-direction , respectively . 
   due to symmetry all other components of the ccf are zero .    as a reference configuration 
   we consider a single spherical colloid of radius @xmath99 with bc @xmath100 at a surface - to - surface distance @xmath101 from a planar substrate with bc @xmath36 . 
   this colloid experiences ( only ) a normal ccf    @xmath102    in the following we normalize the scaling functions @xmath103 and @xmath104 by the amplitude @xmath105 of the ccf acting at @xmath11 on a single colloid for @xmath106 bcs at a surface - to - surface distance @xmath107 . accordingly , in the following we consider the normalized scaling functions    @xmath108    with @xmath109 and @xmath110 . experimentally it can be rather difficult to obtain @xmath105 . 
   a standard alternative way to normalize is to take the more easily accessible amplitude @xmath111 for the ccf at @xmath11 between two parallel plates with @xmath106 bcs , which is given within mft by ( see ref . 
   @xcite and references therein )    @xmath112 ^ 4}{u } \simeq -283.61 u^{-1 } ~,\ ] ]    where @xmath113 is the elliptic integral of the first kind . within mft 
   the amplitude @xmath105 can be expressed in terms of @xmath111 :    @xmath114    equation allows for a practical implementation of the aforementioned normalization , which eliminates the coupling constant @xmath115 , which is unknown within mft . 
   we calculate the normal and lateral forces directly from the numerically determined order parameter profiles @xmath62 by using the stress tensor which , within the ginzburg - landau approach , is given by  @xcite    @xmath116 ~,\ ] ]    with @xmath117 . 
   the first index of the stress tensor denotes the direction of a force , the second index denotes the direction of the normal vector of the surface upon which the force acts . 
   therefore one has    @xmath118    where @xmath119 is a hypersurface enclosing particle @xmath70 , @xmath120 is the @xmath121-th component ( to be summed over ) of its unit outward normal , and @xmath122 is the length of the @xmath123-dimensional hyperaxis of @xmath124 . 
   in particular we focus on the normal and lateral ccfs acting on colloid ( 2 ) for the configuration shown in fig . 
   [ system_sketch ] , for @xmath125 , and with the binary liquid mixture at its critical concentration . in the following analysis 
   we consider colloid ( 1 ) to be fixed in space at a sphere - surface - to - substrate distance @xmath126 , equally sized colloids ( i.e. , @xmath127 ) , and fixed @xmath128 bc for the substrate . 
   we proceed by varying either the vertical ( @xmath74-direction ) or the horizontal ( @xmath42-direction ) position of colloid ( 2 ) by varying either @xmath38 or @xmath129 , respectively . 
   we also consider different sets of bcs for the colloids . in the following results the numerical error is typically less than @xmath130 , unless explicitly stated otherwise . 
   in fig .  [ d2_z ] 
   we show the behavior of the normalized [ eq .  ] scaling function @xmath131 of the normal ccf acting on colloid ( 2 ) with @xmath132 bcs close to a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath133 bc . 
   the scaling functions are shown as functions of the scaling variable ratio @xmath134 , i.e. , for @xmath27 . 
   the various lines correspond to distinct fixed values of @xmath135 as the sphere - surface - to - substrate distance in units of the sphere radius . 
   thus fig . 
   [ d2_z ] shows the temperature dependence of the normal ccf on colloid 2 for three different values of @xmath38 and for colloid ( 1 ) fixed in space .    from fig . 
   [ d2_z ] ( a ) one can see that , for colloid ( 1 ) with @xmath136 bc and for any given value of @xmath137 , the scaling function of the normal ccf acting on colloid ( 2 ) with @xmath138 bc changes sign upon varying @xmath135 . due to the change of sign of @xmath139 , for any value of @xmath137 
   there is a certain value @xmath140 at which the normal ccf acting on colloid ( 2 ) vanishes . for sphere - surface - to - substrate distances sufficiently large such that @xmath141 , colloid ( 2 ) is pushed away from the substrate due to the dominating repulsion between the two colloids in spite of the attraction by the substrate , whereas for @xmath142 it is pulled to the substrate due to the dominating attraction between it and the substrate 
   this implies that , in the absence of additional forces , levitation of colloid ( 2 ) ( i.e. , zero total normal force ) at height @xmath143 is not stable against perturbations of the sphere - surface - to - substrate distance . on the other hand , upon varying temperature , 
   any distance @xmath38 can become a stable levitation position for colloid ( 2 ) with @xmath144 bc in the presence of colloid ( 1 ) with @xmath133 bc [ see fig . 
   [ d2_z ] ( b ) , according to which each scaling function corresponding to a certain value of @xmath38 exhibits a zero so that , at this zero , increasing ( decreasing ) @xmath38 at fixed temperature leads to an attraction ( repulsion ) to ( from ) the substrate ] . in this case 
   the attraction between the two colloids is dominating for large sphere - surface - to - substrate distances @xmath135 , while the repulsion between colloid ( 2 ) and the substrate dominates for small values of @xmath135 .     of the normal ccf acting on colloid ( 2 ) with bcs @xmath145 in ( a ) and @xmath144 in ( b ) . 
   the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath146 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black lines ) , 1.5 ( red lines ) , and 2 ( green lines ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . 
   as expected , the forces become overall weaker upon increasing @xmath135 . 
   @xmath150 @xmath151 implies that the colloid is attracted to ( repelled from ) the substrate along the @xmath74-direction . ] 
   figure  [ d2_x ] shows the behavior of the normalized scaling function @xmath152 of the lateral ccf acting on colloid ( 2 ) in the presence of colloid ( 1 ) having the same bc , i.e. , @xmath153 . 
   the scaling functions are shown as functions of the scaling variable ratio @xmath154 . from figs . 
   [ d2_x ] ( a ) and ( b ) one can infer that @xmath155 . 
   therefore colloid ( 2 ) is always attracted towards colloid ( 1 ) which has the same bc . 
   hence the substrate does not change the sign of the lateral ccf as compared with the attractive lateral ccf in the absence of the confining substrate . 
   however , the shapes of the scaling functions for @xmath156 bcs [ fig .  [ d2_x ] ( a ) ] differ from the ones for @xmath157 bcs [ fig . 
   [ d2_x ] ( b ) ] ; without the substrate , they are identical . in the former case and in contrast to the latter one , the scaling functions exhibit minima above @xmath11 , which is reminiscent of the shape of the corresponding scaling functions in the absence of the substrate .     of the lateral ccf acting on colloid ( 2 ) facing a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath158 bcs . 
   the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath154 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black curves ) , 1.5 ( red curves ) , and 2 ( green curves ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . 
   as expected , the forces become overall weaker upon increasing @xmath135 . 
   @xmath159 implies that colloid ( 2 ) is attracted towards colloid ( 1 ) . 
   two different sets of @xmath160 bcs are considered : @xmath156 in ( a ) and @xmath157 in ( b ) . ]    in fig . 
   [ l_x ] we show the results obtained for the normalized scaling functions @xmath161 of the lateral ccf acting on colloid ( 2 ) . in fig . 
   [ l_x](a ) the scaling function is shown as function of the scaling variable ratio @xmath154 ; the black , red , and green curves correspond to @xmath162 , @xmath163 , and @xmath164 , respectively . in the absence of the substrate , 
   the ccf between two colloids with opposite bcs is repulsive . 
   however , as shown in fig . 
   [ l_x](a ) , in the presence of the substrate with @xmath128 bc , there is a change of sign in the scaling function of the lateral ccf . 
   this implies that the lateral ccf acting between the two colloids changes from being attractive to being repulsive ( or reverse ) upon decreasing ( increasing ) the reduced temperature . 
   thus temperature allows one to control both the strength and the sign of the lateral ccf in the case of two colloids with opposite bcs being near a wall . 
   the at first sight unexpected lateral attraction between two colloids with opposite bcs in the presence of the substrate ( i.e. , @xmath165 ) can be understood as follows . in the absence of the two colloids , the order parameter profile 
   @xmath166 is constant along any path within a plane @xmath167 because in this case @xmath168 . 
   since the substrate area is much larger than the surface areas of the colloids , one can regard the immersion of these colloidal spheres as a perturbation of this profile . in figs . 
   [ l_x](a ) and ( b ) , the region within which the scaling function is negative ( corresponding to an attractive force ) indicates that under these circumstances [ i.e. , when the colloids are sufficiently away from each other ; see fig . 
   [ l_x](b ) ] the perturbation generated by the presence of the spheres decreases upon decreasing the lateral distance between them . 
   this causes the colloids to move towards each other in order to weaken the perturbation by reducing its spatial extension ; this amounts to an attraction , i.e. , @xmath165 . on the other hand , when they are sufficiently close to each other the pairwise interaction between the two colloids dominates and the total lateral ccf is positive ( i.e. , repulsive ) . in fig . 
   [ l_x](c ) we show how the equilibrium lateral distance @xmath169 measured in units of @xmath170 varies as function of temperature , i.e. , @xmath170 for fixed @xmath99 .     of the lateral ccf acting on colloid ( 2 ) . 
   both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . 
   the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . 
   @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . 
   black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig .  [ system_sketch ] ] . 
   the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . 
   ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . 
   each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . 
   the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) 
   we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . 
   note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye 
   the four data points are connected by straight lines.,title="fig : " ]    of the lateral ccf acting on colloid ( 2 ) . 
   both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . 
   the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . 
   @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . 
   black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . 
   [ system_sketch ] ] . 
   the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . 
   ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . 
   each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . 
   the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) 
   we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . 
   note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye 
   the four data points are connected by straight lines.,title="fig : " ]    of the lateral ccf acting on colloid ( 2 ) . 
   both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . 
   the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . 
   @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . 
   black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . 
   [ system_sketch ] ] . 
   the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . 
   ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . 
   each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . 
   the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) 
   we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . 
   note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye 
   the four data points are connected by straight lines.,title="fig : " ]    in order to determine the preferred arrangement of the colloids , we have also analyzed the direction of the total ccf @xmath180 acting on colloid ( 2 ) [ see eq .  ] for several spatial configurations and bcs . for @xmath156 bcs 
   we have found that the colloids tend to aggregate laterally in such a way that several particles with @xmath22 bc , facing a substrate with the same bc , can be expected to form a monolayer on the substrate . on the other hand , for the case of @xmath157 bcs 
   , we have found that the colloids can be expected to aggregate on top of each other so that a collection of colloids with such bcs is expected to form three - dimensional sessile clusters . 
   these tendencies become more pronounced upon approaching the critical point ( see fig . 
   [ direction_resulting_force_d2 ] ) . 
   similar results have been found by soyka et al . 
   @xcite in experiments using chemically patterned substrates . for them 
   the authors have found indeed that colloids with @xmath23 bc distributed over those parts of the substrate with the same bc [ which is equivalent to @xmath156 bcs ] aggregate and form a single layer . moreover , they have found that colloids distributed over parts of the substrate with opposite bc [ corresponding to @xmath157 bcs ] form three - dimensional clusters . 
   and @xmath157 bcs , with @xmath181 . 
   the black rectangles represent the substrate and blue circles represent colloid ( 1 ) while black , red , and green circles represent colloid ( 2 ) with @xmath182 , and 2 , respectively . 
   the centers of all colloids lie in the plane @xmath65 . ] 
   we have determined the many - body force acting on particle @xmath183 by subtracting from the total force @xmath184 [ see eq .  ] 
   the sum of the pairwise forces acting on it , i.e. , the colloid - colloid ( cc ) and the colloid - substrate ( cs ) forces . 
   accordingly the many - body ccf @xmath185 acting on colloid ( @xmath70 ) is given by ( see fig .  [ system_sketch ] )    @xmath186    where    @xmath187    with    @xmath188    is the pairwise colloid - colloid force ( acting on colloid 2 @xmath23 or 1 @xmath22 with 2 having the larger @xmath42-coordinate ) expressed in terms of the absolute value @xmath189 of the force between two colloids at surface - to - surface distance @xmath190 in free space . 
   @xmath191 is the ccf between the substrate and a single colloid @xmath183 . 
   we have studied both the normal    @xmath192    and the lateral    @xmath193    many - body ccfs which are characterized by corresponding scaling functions [ compare eqs .   and ] :    @xmath194    and    @xmath195    in fig . 
   [ mb_l_z ] we show the normalized [ see eqs .   and ] scaling functions @xmath196 of the many - body normal ccf acting on colloid ( 2 ) . 
   this figure reveals similar results for @xmath156 [ fig . 
   [ mb_l_z ] ( a ) ] and @xmath197 [ fig .  [ mb_l_z ] ( b ) ] bcs . in these cases , 
   each mb scaling function exhibits both a maximum and at least one minimum , the former one appearing for smaller values of the scaling variable @xmath198 ( i.e. , at temperatures closer to @xmath11 ) . for a certain range of temperatures close to @xmath11 , as the distance @xmath129 between the colloids increases , the many - body normal ccf changes from attractive to repulsive . 
   this shows that for each temperature within this range there is a lateral distance @xmath199 for which the many - body contribution to the normal force acting on colloid ( 2 ) is zero . 
   this means that under such conditions the sum of pairwise forces provides a quantitatively reliable description of the total force acting on colloid ( 2 ) . for temperatures sufficiently far from @xmath11 , the many - body normal ccf is always attractive with a monotonic dependence on @xmath198 . here , as in figs . 
   [ d2_z ] - [ l_x ] , the ccfs decay exponentially for @xmath200 . 
   as expected , the many - body effects are more pronounced if the colloids are closer to each other and/or closer to the substrate . 
   indeed for situations in which the colloids are close to each other [ see , e.g. , the black curves in figs . 
   [ mb_l_z ] ( a ) and ( b ) ] we have found that when the normal many - body ccf reaches its maximal strength , corresponding to the minimum of the scaling function @xmath201 at @xmath202 , the relative contribution of the many - body ccf reaches @xmath0 of the strength of the total normal ccf . for larger distances between the colloids [ see , e.g. , the green curves in figs 
   .  [ mb_l_z ] ( a ) and ( b ) ] this relative contribution is smaller ( around @xmath203 for @xmath204 and @xmath205 for @xmath206 ) .     of the many - body normal ccf acting on colloid ( 2 ) for @xmath207 and @xmath208 . 
   the scaling functions are shown as functions of the scaling variable ratio @xmath209 for the sets of bcs @xmath156 in ( a ) and @xmath197 in ( b ) . 
   the black , red , and green lines correspond to @xmath210 , and @xmath211 , respectively . 
   figures [ d2_z](a ) and [ mb_l_z](b ) allow a direct comparison between the full ccf and the corresponding many - body contribution ( note the different scales of the ordinates . ) ]    we are not aware of results for the quantum - electrodynamic casimir interactions which are obtained along the same lines as our ccf analysis above . 
   nonetheless , in order to assess the significance of our results we compare them with the results in ref . 
   @xcite , which is the closest comparable study which we have found in the literature . 
   therein the authors study theoretically two dielectric spheres immersed in ethanol while facing a plate . depending on the kind of fluid and on the materials of the spheres and of the plate as well as on the distances involved , 
   also the quantum - electrodynamic casimir force can be either attractive or repulsive . 
   it is well known  @xcite that for two parallel plates with permittivities @xmath212 and @xmath213 separated by a fluid with permittivity @xmath214 and without further boundaries , the quantum - electrodynamic casimir force is repulsive if @xmath215 within a suitable frequency range . in ref .  @xcite it is stated that this also holds for two spheres immersed in a fluid . 
   the authors of ref . 
   @xcite analyze the effect of nonadditivity for the above system by studying the influence of an additional , adjacent substrate on the equilibrium separation @xmath56 between two nanometer size dielectric spheres . 
   to this end , they consider two spheres of different materials with the same radii @xmath207 and the same surface - to - plate distances @xmath216 and analyze how the lateral equilibrium distance @xmath217 between the spheres depends on @xmath101 . by comparing the equilibrium distance @xmath217 with that in the absence of the substrate , @xmath218 
   , they find that @xmath217 increases or decreases ( depending on the kind of materials of the spheres ) by as much as @xmath219 as the distance from the plate varies between @xmath220 and @xmath221 . 
   they also find that `` the sphere - plate interaction changes the sphere - sphere interaction with the same sign as @xmath101 becomes smaller '' , which means that if the sphere - plate force is repulsive ( attractive ) , the sphere - sphere one will become more repulsive ( attractive ) upon decreasing the distance from the plate @xmath101 . by construction 
   these changes are genuine many - body contributions . in the case of two chemically different spheres , the sign of the many - body force contribution ( i.e. , whether it is attractive or repulsive ) agrees with the sign of the stronger one of the two individual sphere - plate interactions .    in fig . 
   [ comparing ] we show schematically the system considered in ref . 
   @xcite [ ( a ) and ( b ) ] and the system considered here [ ( c ) , ( d ) , and ( e ) ] . for the quantum - electrodynamic casimir effect , 
   the dielectric spheres are represented by circles of equal radii , with the green one corresponding to a polystyrene sphere and the red one to a silicon sphere . 
   the semi - infinite plates are represented by gray and yellow rectangles for teflon and gold , respectively . 
   the whole configuration is immersed in ethanol which , for simplicity , is not shown in the figure . 
   the dashed arrows indicate the direction of the strongest of the two pairwise sphere - substrate forces , while the solid arrows indicate the direction of the lateral many - body force . 
   the directions of the arrows in figs . 
   [ comparing ] ( a ) and ( b ) are chosen as to illustrate the findings in ref . 
   @xcite , according to which the sign of the _ lateral _ many - body force is the same as the one of the strongest _ normal _ pairwise sphere - plate force : attractive in ( a ) and repulsive in ( b ) . 
   also in the case of the critical casimir forces , depicted in figs .  [ comparing ] ( c ) , ( d ) , and ( e ) , 
   we represent the colloids by circles and the laterally homogeneous semi - infinite substrate by rectangles . 
   the orange filling represents the @xmath22 bc while the blue filling represents the @xmath23 bc . 
   again , the dashed arrows indicate the direction of the stronger one of the two pairwise ( normal ) colloid - substrate forces , while the solid arrows indicate the direction of the lateral many - body contribution to the ccf . as one can infer from fig . 
   [ mb_x ] , the _ lateral _ many - body ccf acting on colloid ( 2 ) is always attractive for the given geometrical configuration , regardless of the bcs .     and alluding to the system 
   studied in ref . 
   @xcite , the circles represent the projections of dielectric spheres with equal radii , the green one corresponding to polystyrene and the red one to silicon ; the rectangles represent semi - infinite plates with their surfaces perpendicular to the @xmath222 plane , the gray and the yellow one being teflon and gold , respectively . 
   the system is immersed in ethanol , which is not indicated in the figure . in ( a ) and ( b ) 
   each dashed arrow indicates the direction of the stronger one of the two corresponding pairwise forces between the dielectric spheres and the plate , which turns out to determine the direction of the many - body lateral force acting on the spheres : if the stronger one of the two pairwise forces is attractive [ repulsive ] , the lateral many - body force will also be attractive [ repulsive ] ( see ref . 
   also in the case of the critical casimir interaction [ ( c ) , ( d ) , and ( e ) ] , the circles and rectangles represent projections of spherical colloids and of homogeneous substrates , respectively : orange and blue indicate @xmath22 and @xmath23 bcs , respectively . in ( c ) and 
   ( d ) the pairwise normal forces between each of the two spheres and the substrate are equal : attractive in ( c ) and repulsive in ( d ) . in ( e ) the two pairwise 
   normal forces have opposite directions with the repulsive one being the stronger one  @xcite . 
   the corresponding dashed arrows have the same meaning as in ( a ) and ( b ) . according to fig . 
   [ mb_x ] the many - body lateral ccfs are attractive for all three cases ( c ) , ( d ) , and ( e ) . 
   the comparison shows that the systems in ( a ) and ( c ) behave similarly . 
   however , the behavior of system ( b ) has no counterpart for ccfs [ see ( d ) and ( e ) ] . in this figure 
   all surface - to - surface distances equal the sphere radius , which in our notation corresponds to @xmath223 . ] 
   we can also compare our results with those for two atoms close to the surface of a planar solid body . 
   mclachlan  @xcite has tackled this problem by treating the solid as a uniform dielectric . by using the image method he derived an expression for the many - body corrections to the pairwise interaction energies , i.e. , the atom - atom ( london ) and the atom - surface energies , in order to obtain the total interaction energy between the two atoms close to the surface . 
   qualitatively , he found that the leading contribution of the many - body correction leads to a repulsion if the atoms are side by side , i.e. , at equal surface - to - substrate distances . 
   rauber et al . 
   @xcite used mclachlan s approach to study the electrodynamic screening of the van der waals interaction between adsorbed atoms and molecules and a substrate . 
   the latter plays a role which is `` analogous to that of the third body in the three - body interaction between two particles embedded in a three - dimensional medium '' . 
   the van der waals interaction between the two atoms at equal distances from the substrate is altered by the presence of the solid substrate and this perturbation is given by  @xcite    @xmath224    where @xmath225 is the distance between the atoms , @xmath121 is the height above the image plane , which is the same for both atoms , and @xmath226 . 
   the coefficients are given by    @xmath227    and    @xmath228    with    @xmath229 / \left [ \epsilon(i\zeta ) + 1 \right ] ~,\ ] ]    where @xmath230 is an imaginary frequency , @xmath231 is the polarizability of the atoms ( with the dimension of a volume ) , and @xmath232 is the dielectric function of the solid ( i.e. , the substrate ) . the lateral force due to the perturbation potential given by eq .  , 
   which is the analogue of the many - body contribution to the lateral ccf , follows from differentiating @xmath233 with respect to @xmath225 :    @xmath234 ~.\end{gathered}\ ] ]    in figure  [ mclachan_fig ] we plot the lateral force given by eq .   as function of the distance @xmath225 between the two atoms for several ( equal ) distances @xmath121 above the substrate . 
   we use the values provided in ref . 
   @xcite for the coefficients @xmath235 and @xmath236 : @xmath237 ev@xmath238()@xmath239 and @xmath240 ev@xmath238()@xmath239 , which correspond to ne , and @xmath241 ev@xmath238()@xmath239 and @xmath242 ev@xmath238()@xmath239 , which correspond to ar . as one can infer from fig . 
   [ mclachan_fig ] , the many - body contribution to the lateral van der waals force is always repulsive and , as the two atoms approach the substrate , its strength increases . on the other hand , in the case of the many - body contribution to the lateral ccf , we have found that it is attractive for all bcs considered , if the surface - to - surface distances between the spheres and the sphere - surface - to - substrate distances are equal to each other and to the radius of the spheres ( see fig .  [ mb_x ] ) . 
   further , we can _ quantitatively _ compare our results with those from refs . 
   @xcite and  @xcite . to this end 
   , we assign values to the geometrical parameters characterizing the configuration of the two atoms close to the substrate and compare the results of refs . 
   @xcite and  @xcite with those for similar configurations in our model . 
   for example , estimating the many - body contribution to the lateral van der waals force for a configuration of two atoms close to a substrate corresponding to the configuration associated with the black curve   the black curve corresponds to a function of temperature but there is no temperature dependence of the van der waals force between the atoms . 
   therefore the comparison has to be carried out by choosing a certain value of @xmath243 . in fig . 
   [ mb_x ] ( i.e. , @xmath244 in the case of the ccf and @xmath245 in the case of the two atoms ) , one obtains from eq .   a value for the relative contribution of the many - body force to the lateral force which corresponds to ca . 
   this is comparable with the relative contribution of the many - body force to the lateral ccf sufficiently close to @xmath11 , although in the case of the van der waals force it is repulsive ( fig . 
   [ mclachan_fig ] ) while in the case of the ccf ( fig . 
   [ mb_x ] ) it is attractive .    considering the decay of the many - body contribution to the normal ccf as function of the surface - to - surface distance @xmath129 between the spheres ( for @xmath216 ) , we can compare the corresponding decay of the normal many - body force @xmath246 given by the potential @xmath233 in eq .  . 
   for small separations @xmath225 between the atoms the many - body contribution to the normal van der waals force increases as @xmath247 , while for large separations it decays as @xmath248 . by analyzing the data shown in fig . 
   [ mb_l_z ] one finds that for @xmath249 the scaling function of the many - body contribution to the normal ccf decays slower than @xmath250 . 
   this means that in this temperature regime the many - body contribution to the normal ccf is much more long ranged than the corresponding contribution to the normal van der waals force in the case of two atoms close to a surface . for fixed @xmath225 the many - body contribution to the lateral van der waals force decays as @xmath251 upon increasing the distance of both atoms from the substrate whereas the normal force on a single atom decays as @xmath252 .    as a final remark we point out that we have not found a completely stable configuration for the two colloids ( fig . 
   [ system_sketch ] ) : whenever there is a stable position in the horizontal ( vertical ) direction , the force is nonzero in the vertical ( horizontal ) direction . 
   for example , consider a vertical path with @xmath253 in figs . 
   [ d2_z](b ) and [ d2_x](b ) . from the first one 
   can see that along this path the normal ccf changes from being repulsive to being attractive as the sphere - surface - to - substrate distance for colloid ( 2 ) is increased , implying that there is a vertical position of colloid ( 2 ) in which the normal ccf is zero . 
   however , according to fig . 
   [ d2_x](b ) the lateral ccf is always attractive regardless of the vertical position of colloid ( 2 ) . 
   this means that there is a configuration which is stable only in the normal direction . 
   accordingly , a dumbbell configuration with a rigid thin fiber between the two colloids can levitate over the substrate . 
   whether this configuration is stable with respect to a vertical tilt remains as an open question .     of the lateral many - body ccf acting on colloid ( 2 ) for @xmath207 and @xmath254 . 
   the scaling function is shown as function of the scaling variable ratio @xmath255 . 
   the black , red , and green lines correspond to the bcs @xmath157 , @xmath197 , and @xmath156 , respectively . 
   for all three bcs the many - body contribution is not monotonic as function of temperature . 
   quantitatively the green , red , and black curves here should be compared with the black curves in figs . 
   [ d2_x](a ) , [ d2_x](b ) , and [ l_x](a ) , respectively . 
   however , for the data shown in _ this _ figure the error bars ( not shown ) due to limits of the numerical accuracy are between 10@xmath256 and 15@xmath256 . 
   this is the main reason why we refrain from showing what would be an instructive plot such as @xmath257 as a function of @xmath258 for various values of @xmath259 and @xmath170 ( as we did in fig . 
   [ l_x ] ) , which would allow for a direct comparison with the case of atoms . ]    , derived from the expression for the excess potential given in refs . 
   @xcite and  @xcite [ eq .  ] . 
   the forces are plotted as functions of the lateral separation @xmath225 between the two atoms for several equal vertical distances @xmath121 of the atoms from the substrate . 
   the curves correspond to two sets of values for the coefficients @xmath235 and @xmath236 in eq . 
   @xcite , corresponding to ne ( solid lines ) and ar ( dashed lines ) . ] 
   we now turn our attention to the case in which the colloids are vertically aligned with respect to a planar , homogeneous substrate , i.e. , when their centers have the same coordinates in both the @xmath42 and the @xmath260 direction ( see fig .  [ vertical_sketch ] ) . 
   we focus on the normal ccf acting on colloid ( 1 ) when the system is immersed in a near - critical binary liquid mixture at its critical concentration . as before we consider @xmath31 bcs corresponding to a strong adsorption preference for one of the two components of the confined liquid . 
   in particular , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a laterally homogeneous substrate with bc @xmath36 . 
   colloid ( 1 ) is positioned at a sphere - surface - to - substrate distance @xmath37 and colloid ( 2 ) is at a surface - to - surface distance @xmath261 from colloid ( 1 ) ( see fig . 
   [ vertical_sketch ] ) . 
   the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath262 and @xmath263 so that the distance between the centers , along the @xmath74-axis , is given by @xmath264 . as before , the bcs of the system as a whole are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . 
   and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the colloidal particle ( 1 ) with bc @xmath34 
   is located vertically at the sphere - surface - to - substrate distance @xmath37 , whereas the colloidal particle ( 2 ) with bc @xmath64 is located vertically at the surface - to - surface distance @xmath261 between the spheres . 
   the substrate exhibits bc @xmath36 . 
   the vertical distance between the centers of the spheres along the @xmath74-direction is given by @xmath264 , while the centers of both spheres lie on the vertical axis @xmath265 . 
   in the case of four spatial dimensions the figure shows a three - dimensional cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 [ see eq .  ] . ] 
   the normal ccf @xmath266 acting on colloid ( 1 ) along the @xmath74-direction takes the scaling form    @xmath267    where @xmath268 ( i.e. , @xmath269 for @xmath27 and @xmath270 for @xmath10 ) , @xmath76 , @xmath271 , and @xmath272 ; @xmath273 . 
   equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . 
   @xmath274 is the force on a hypercylinder divided by its extension in the translationally invariant direction [ see eq .  ] . 
   we use the same reference system as the one described by eq .   in order to normalize the scaling function defined in eq .   according to    @xmath275 
   we calculate the many - body contribution to the normal ccf acting on particle ( 1 ) @xmath276 by subtracting from the total force the sum of the pairwise forces acting on it , i.e. , the colloid - colloid and the colloid - substrate forces [ see eq .  ] . 
   this many - body force takes the scaling form    @xmath277    in fig . 
   [ mb_vertical_z ] we show the normalized [ see eqs .  , , and ] scaling functions @xmath278 of the many - body contribution to the normal ccf acting on colloid ( 1 ) as functions of the scaling variable ratio @xmath279 for two spherical colloids of the same size ( @xmath207 ) . keeping the surface - to - surface distance between the spheres fixed at @xmath280 , we vary the sphere - surface - to - substrate distance @xmath37 for several bcs : @xmath156 in ( a ) , @xmath281 in ( b ) , @xmath197 in ( c ) , and @xmath157 in ( d ) .    from figs . 
   [ mb_vertical_z](a ) and ( d ) one can infer that if the colloids have symmetric bcs , the scaling function of the many - body normal ccf acting on colloid ( 1 ) is negative ( i.e. , it is directed towards the substrate ) for any value of @xmath198 . on the other hand , for non - symmetric bcs between the colloids [ figs . 
   [ mb_vertical_z](b ) and ( c ) ] , the many - body contribution to the normal ccf acting on colloid ( 1 ) is positive for any value of @xmath198 . 
   the apparent change of sign in figs .  [ mb_vertical_z](b ) and ( c ) is likely to be an artifact occurring within the error bars due to numerical imprecision . 
   the relative contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . 
   we point out that this configuration with the the two colloids vertically aligned with respect to the substrate allows for a wide range of interesting aspects which will be further explored in future works . 
   we have investigated critical casimir forces ( ccfs ) for a system composed of two equally sized spherical colloids ( @xmath207 ) immersed in a near - critical binary liquid mixture and close to a laterally homogeneous substrate with @xmath128 boundary condition ( bc ) ( see fig .  [ system_sketch ] ) . by denoting the set of bcs of the system as @xmath44 , where @xmath282 corresponds to the bc at colloid @xmath283 and @xmath46 to the bc at the substrate , we have first focused on the total normal and lateral forces acting on one of the colloids [ labeled as `` colloid ( 2 ) '' ] for several geometrical configurations of the system and various combinations of bcs at the colloids . 
   both the normal and the lateral forces are characterized by universal scaling functions [ eqs .   and , respectively ] , which have been studied in the one - phase region of the solvent as functions of @xmath284 and @xmath285 . 
   @xmath129 is the surface - to - surface distance between the two colloids , and @xmath178 is the bulk correlation length of the binary mixture in the mixed phase . we have used mean - field theory together with a finite element method in order to calculate the order parameter profiles , from which the stress tensor renders the normalized scaling functions associated with the ccfs .    for the scaling function of the total normal ccf acting on colloid ( 2 ) with @xmath145 bc , in the presence of colloid ( 1 ) with @xmath286 , we have found ( fig .  [ d2_z](a ) ) that the scaling function changes sign for a fixed value of @xmath284 as the distance @xmath38 between colloid ( 2 ) and the substrate increases , signaling the occurrence of an unstable mechanical equilibrium configuration of a vanishing normal force . for the total normal ccf acting on colloid ( 2 ) with @xmath144 bc and in the presence of colloid ( 1 ) with @xmath133 , we have found ( fig .  [ d2_z](b ) ) 
   that the force changes sign upon changing the temperature . 
   for this combination of bcs , the equilibrium configuration of colloid ( 2 ) is stable in the normal direction .    without a substrate , at the critical composition of the solvent 
   the ccf between two @xmath22 spheres is identical to the one between two @xmath23 ones . 
   this degeneracy is lifted by the presence of the substrate as one can infer from the comparison of the scaling functions for the lateral ccfs for @xmath156 and @xmath157 bcs [ see figs . 
   [ d2_x](a ) and ( b ) , respectively ] . in the first case 
   , the shape of the scaling function resembles that of the two colloids far away from the substrate , with a minimum at @xmath287 . in the second case 
   this minimum at @xmath288 disappears . 
   these substrate - induced changes are more pronounced if the two spheres are close to the substrate ( fig . 
   [ d2_x ] ) . without a substrate 
   the ccf between spheres of opposite bcs is purely repulsive . in the presence of a substrate 
   the corresponding lateral ccf for @xmath197 bcs turns attractive for large lateral distances @xmath129 ( fig . 
   [ l_x ] ) , which is a pure many - body effect . 
   we have also studied the direction of the total ccf acting on colloid ( 2 ) for various spatial configurations and bcs in order to assess the preferred arrangement of the colloids . for @xmath156 bcs 
   we have found that they tend to aggregate laterally . in this case a collection of colloids with @xmath22 bcs , facing a substrate with the same bc , are expected to form a monolayer on the substrate . 
   for the situation of @xmath157 bcs , we have found that the colloids are expected to aggregate on top of each other . 
   this indicates that a set of several colloids with such bcs are expected to form three - dimensional clusters . 
   these tendencies are enhanced upon approaching the critical point ( fig . 
   [ direction_resulting_force_d2 ] ) .    by calculating the pairwise colloid - colloid and colloid - substrate forces and subtracting them from the total force , 
   we have determined the pure many - body contribution to the force acting on colloid ( 2 ) . for the scaling functions associated with the normal many - body ccfs we have found the interesting feature of a change of sign at fixed temperature upon varying the lateral position of colloid ( 2 ) ( fig . 
   [ mb_l_z ] ) . 
   this implies that , for a given temperature , there is a lateral position where the normal many - body ccf is zero , in which case the sum of pairwise forces provides a quantitatively reliable description of the interactions of the system . as expected we have found that the contribution of the many - body ccfs to the total force is large if the colloid - colloid and colloid - substrate distances are small , as well as if the binary liquid mixture is close to its critical point . 
   we have compared our results with corresponding ones for quantum - electrodynamic casimir interactions . to this end 
   we have referred to the results in ref . 
   @xcite for two dielectric spheres immersed in ethanol and facing a plate . 
   these authors analyze the influence of the distance @xmath289 from the plate on the equilibrium separation @xmath217 between the spheres , which are subject to quantum - electrodynamic casimir forces . 
   they find that the lateral many - body force is attractive ( repulsive ) if the stronger one of the two normal sphere - plate forces is attractive ( repulsive ) [ figs . 
   [ comparing](a ) and ( b ) ] . on the other hand , 
   in the case of ccfs we have found that for a configuration in which the surface - to - surface distance between the colloids is equal to the sphere - surface - to - substrate ones and equal to the radius of the spheres , the many - body contribution to the lateral ccf is always attractive , regardless of the bcs ( fig . 
   [ mb_x ] ) . 
   we have also compared our results with the corresponding ones for the case of two atoms close to the planar surface of a solid body . in this respect 
   we have referred to the mclachlan model  @xcite for the many - body contribution to the van der waals potential [ see eq .   and fig . 
   [ mclachan_fig ] ] and the results from ref . 
   @xcite . from this comparison 
   we have found that if the two atoms are fixed at the same distance from the surface of the solid body , the normal many - body contribution to the total van der waals force decays with the atom - atom distance @xmath225 as @xmath248 for large atom - atom distances . 
   this decay is much faster than the decay we estimate for the many - body contribution to the normal ccf , which within a suitable range appears to be slower than @xmath250 . 
   furthermore , we have found that the many - body contribution to the lateral van der waals force is repulsive while the corresponding many - body ccf is attractive regardless of the set of bcs . 
   finally we have considered the configuration in which the two colloids are vertically aligned with respect to the substrate ( fig . 
   [ vertical_sketch ] ) . 
   we have calculated the many - body contribution to the normal ccf acting on colloid ( 1 ) for two spherical colloids of the same size ( @xmath207 ) keeping the sphere - surface - to - surface distance @xmath280 fixed ( fig . 
   [ mb_vertical_z ] ) . 
   we have varied the sphere - surface - to - substrate distance @xmath37 for several bcs and have found that if the colloids have the same bcs , the many - body contribution to the normal ccf is directed towards the substrate [ figs . 
   [ mb_vertical_z](a ) and ( d ) ] , whereas for colloids with opposite bcs , the many - body contribution to the normal ccf is directed away from the substrate [ figs . 
   [ mb_vertical_z](b ) and ( c ) ] . 
   we have found that the contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . 
   t.g.m . would like to thank s. kondrat for valuable support with the computational tools used to perform the numerical calculations . 
   s.d . thanks m. cole for providing ref .  @xcite .
summary: within mean - field theory we calculate the scaling functions associated with critical casimir forces for a system consisting of two spherical colloids immersed in a binary liquid mixture near its consolute point and facing a planar , homogeneous substrate . for several geometrical arrangements and boundary conditions we analyze the normal and the lateral critical casimir forces acting on one of the two colloids . 
   we find interesting features such as a change of sign of these forces upon varying either the position of one of the colloids or the temperature . by subtracting the pairwise forces from the total force we are able to determine the many - body forces acting on one of the colloids . 
   we have found that the many - body contribution to the total critical casimir force is more pronounced for small colloid - colloid and colloid - substrate distances , as well as for temperatures close to criticality , where the many - body contribution to the total force can reach up to @xmath0 .