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colloidal particles dissolved in a nematic solvent experience long - ranged interactions  @xcite .
these effective interactions are generated by the distortions of the liquid crystal director around the particles and result in their clustering and self - organization  @xcite . upon approaching and at the nematic - isotropic transition , the solvent forms nematic and isotropic domains , which leads to an additional interaction of the particles with the interface between the two phases  @xcite .
the resultant morphology of the network formed by the particles in colloid - liquid crystal composites is very sensitive to this interaction  @xcite .
in fact , colloidal particles can be captured by the nematic - isotropic interface and even dragged by a _ moving _
interface  @xcite .
this enables one to manipulate tiny particles in suspension , which in a more general context is of technological importance for manufacturing of @xmath0-papers and electrophoretic displays  @xcite , separation of bacterial species and living cells  @xcite , trapping of dna and polymer particles  @xcite , as well as growth of photonic crystals  @xcite .
the structures of self - assembled colloid layers at _ free _ surfaces of nematic films can be tuned by their thickness  @xcite .
recent experiments of dragging colloidal particles by nematic - isotropic interfaces  @xcite demonstrated that a single colloidal particle is attracted by the interface .
this conclusion is in accordance with theoretical results obtained by minimizing the corresponding landau - de gennes free energy  @xcite .
the resulting force on the particle is non - monotonic and is roughly proportional to the particle radius .
experimental results  @xcite also suggested that the particles captured by the interface interact with each other differently than in the pure nematic or isotropic phase , e.g. , the particle - particle interaction depends on the interfacial curvature ; particles which interact only weakly in the bulk can segregate and form two - dimensional clusters once they are captured by the interface ( see fig .
[ fig : experiment ] ) .
the appearance of additional interface induced features in the interaction between colloidal particles can be expected from analogous phenomena at interfaces of simple fluids  @xcite which , however , still pose unresolved challenges in understanding  @xcite . in liquid crystals the situation is even more complicated .
the director deformations extend into the nematic phase , i.  e. , in addition to interfacial energies , the bulk elasticity contributes to the total free energy of the system .
the effective surface tension ( provided it can be introduced in the first place ) depends on the orientation of the director , which in turn varies along the interface .
the boundary conditions for the director at the particle surfaces give rise to topological defects accompanying the particles . in order to minimize the free energy associated with the director distortions as well as the free energy of the defect cores , the defects often merge with the isotropic phase  @xcite or with each other .
the combination of the above - mentioned effects enriches the phenomena significantly but complicates their theoretical understanding enormously : expressions for the director field , the free energy , or force - distance profiles can hardly be obtained analytically .
numerical calculations are also not straightforward because the interfacial width and the size of the defect cores are normally much smaller that the size of the colloidal particle , so that there are several rather different length - scales involved
. therefore significant computational efforts involving finite - element methods with adaptive meshes are required to address even static problems  @xcite .    in the following
we study the interaction of two long parallel cylindrical colloids trapped at a ( initially ) flat nematic - isotropic interface .
we find that their effective interaction is rather complex : the force - distance profiles can exhibit jumps and hysteresis upon approach or separation .
the particles can either repel or attract each other depending on the material parameters and the geometry .
finally , we propose a simple effective pair potential which can be exploited to study clustering and self - organization of colloids at ni interfaces .
two identical colloidal particles , each of which we take to be a long cylinder of radius @xmath1 with the symmetry axis parallel to the @xmath2 axis , are immersed into a nematic liquid crystal at a separation @xmath3 , see fig .
[ fig : geometry ] .
the nematic order parameter tensor at the boundaries is fixed in a such a way that without the colloids a flat nematic - isotropic interface is formed at @xmath4 .
the order parameter at the top wall is fixed to zero ; the absolute value of the order parameter at the bottom wall is fixed to the bulk order parameter of the nematic phase at two - phase coexistence .
the director orientation at the bottom wall is also fixed and corresponds to the preferred director anchoring at the nematic - isotropic interface , i.e. , parallel or perpendicular to the interface .
the axes of the particles are positioned at @xmath5 . in the course of the free energy minimization @xmath6 adjusts itself such that that the @xmath7component of the force exerted on the particles vanishes .
the system is characterized by the landau - de gennes free energy  @xcite @xmath8 \label{eq : nem_free_en},\end{aligned}\ ] ] where summation over repeated indices is implied and the comma indicates the spatial derivative . the positive constants @xmath9 are assumed to be temperature independent , and @xmath10 is the supercooling temperature of the isotropic phase .
the constants @xmath11 and @xmath12 are related to the frank - oseen elastic constants @xmath13 and @xmath14 and @xmath15 is the bulk nematic order parameter .
the sign of @xmath12 determines the preferred orientation of the director at the nematic - isotropic interface .
@xmath16 ( @xmath17 ) favors planar ( perpendicular ) anchoring .
we introduce the dimensionless temperature @xmath18 .
the bulk nematic phase is stable for @xmath19 with a degree of orientational order given by @xmath20 ; @xmath21 .
we consider both rigid homeotropic and finite anchoring boundary conditions at the colloidal surfaces . in the latter case
the surface free energy density is taken as  @xcite @xmath22 where @xmath23 is the anchoring coefficient and @xmath24 is the preferred tensor order parameter at the colloid surface with @xmath25 as the unit vector along the easy axis direction .    for the constants entering the free energy density ( eq .
( [ eq : nem_free_en ] ) ) we use typical values for a nematic compound 5cb  @xcite : @xmath26 , @xmath27 , @xmath28 , @xmath29 , @xmath30 .
the choices for @xmath12 will be given below .
the nematic - isotropic transition temperature for 5cb is @xmath31 . at coexistence the nematic coherence length (
i.e. , the thickness of the nematic - isotropic interface ) is @xmath32 which sets the smallest length - scale of our description .
the equilibrium distribution of the tensor order parameter @xmath33 is obtained by minimizing the free energy functional  ( [ eq : nem_free_en ] ) numerically using finite elements with adaptive meshing .
the area @xmath34 is triangulated and the functions @xmath33 are linearly interpolated within each triangle .
for this case we chose the anisotropy of the elastic constants as @xmath35 , favoring director alignment parallel to the ni interface and in the plane @xmath36 .
we use rigid homeotropic boundary conditions at the colloidal surfaces . upon minimizing the free energy we have identified two stable orientational configurations . at sufficiently large separations
@xmath3 both colloids are accompanied by a defect of strength @xmath37 ( fig .
[ fig : config_planar](b ) ) .
if the particles move closer , the two defects merge and form a single , but rather extended , defect positioned between the colloids ( fig .
[ fig : config_planar](a ) ) .
we have also observed three metastable configurations .
here either one of the defects , fig .
[ fig : config_planar](c ) , or even both defects , fig .
[ fig : config_planar](d ) and ( e ) , merge with the isotropic phase : the system is trying to reduce its free energy by annihilating the defects .
however , the interface has to bend and almost wrap the particles in order to reach the defects .
this of course increases the interfacial energy , which is roughly proportional to the area of the interface . for small separations , among the metastable configurations only the one shown in fig .
[ fig : config_planar](c ) is possible : due to strong anchoring at the particle surfaces the nematic phase forms a bridge between the particles , preventing the isotropic phase from reaching also the second defect . for larger separations ,
the isotropic phase can extend partially around the colloids and merge with the second defect .
this can occur asymmetrically , as shown in fig .
[ fig : config_planar](d ) , or symmetrically , see fig .
[ fig : config_planar](e ) .
the effective pair potential @xmath38 defined as the free energy per length of the colloidal particles is presented in fig .  [ fig : energy_planar ] . in all cases
the particles repel each other . the branch of @xmath38 which corresponds to the _ stable _ configurations , reveals practically no breaks in slope , even though at @xmath39 the director field around the particles changes abruptly between the configurations ( a ) and ( b ) .
however , if one of the metastable configurations is present in the system , transitions between different branches can occur giving rise to jumps and hysteresis in the force - distance curves upon particle approach / separation .
the choice @xmath40 for the anisotropy of the elastic constants favors director alignment perpendicular to the ni interface in the plane @xmath36 . in this case
we have found only two , stable configurations .
the first one , containing a defect , is shown in fig .
[ fig : config_hom](a )
. it occurs at small separations , where the nematic bridge between the particles forces the director to be parallel to the interface .
the other configuration has no defects , because the nematic bridge with director orientation parallel to the interface disappears and the defect merges with the isotropic phase .
the corresponding effective potential @xmath38 is shown in fig .
[ fig : energy_hom ] . as in the previous case ,
the particles repel each other at small separations where there are large director distortions in the inter - particle region .
however , contrary to the case with planar anchoring , there is a strong attraction between the particles for @xmath41 , where the configuration with the defect is stable .
the branch without the defect results in weak repulsion , as for planar anchoring .
minimization of the free energy is a computationally expensive task , which can not be used , for example , to calculate the forces on the particles each time their positions are advanced using molecular dynamics or monte - carlo algorithms for their kinetics .
use of effective potentials is a typical solution to this problem . in order to find such an appropriate potential
we have calculated the free energy per unit length for particles of different radii , @xmath42 , corresponding to @xmath43 .
since colloids are rather small , the finite director anchoring at the particle surfaces has to be taken into account .
we have used a value appropriate for 5cb on a polymer surface  @xcite , @xmath44 ; for this energy the dimensionless anchoring parameter @xmath45 is of the order of @xmath46 .
minimization results for both planar and homeotropic anchoring of the director at the interface are shown in fig .
[ fig : energy_fit ] ( a ) and ( b ) .
the strong repulsion close to contact ( @xmath47 ) has become smoother for planar and disappeared for homeotropic anchoring at the interface , due to the finite director anchoring at the colloid surfaces .
it turns out that in both cases the following shape of the potential fits the numerical data very well : @xmath48 , & 0 \le d \le d_c \\ u(\infty ) + u_0 \bigl ( \exp   \left [ -\alpha \left ( { d}/{\xi }   \right)^{-\beta } \right ] - 1\bigr ) , & d >
d_c , \end{array } \right . \label{eq : fit}\ ] ] where @xmath49 are ( positive ) fitting parameters .
the first part of the potential ( @xmath50 ) describes a simple harmonic - type attraction between the particles , which we observed for the _ homeotropic _ anchoring ; for _ planar _ interfacial anchoring @xmath51 . in the case of homeotropic anchoring @xmath52
is a critical distance at which the defect annihilates ( see figs .
[ fig : config_hom ] and [ fig : energy_hom ] ) .
we have no analytical support for the functional form of the effective potential ( [ eq : fit ] ) .
however , qualitatively , the effective interaction of the colloidal particles in a bulk nematic phase is known to exhibit a power - law decay at large distances  @xcite . in our case ,
the leading term of the expansion for @xmath53 is indeed @xmath54 . on the other hand , if two colloids are sufficientlly close , so that the regions where the _ order parameter _ changes from their surface to the bulk value start to overlap , and where the defects ( if present ) interact with each other and the particles  @xcite , the functional form of the potential changes . normally , the distortions of the order parameter relax exponentially ( similar to the variations of the order parameter through the nematic - isotropic interface ) resulting in an exponentially varying strength of the interaction .
this is also captured qualitatively by the potential given in eq .
( [ eq : fit ] ) .    the fitting curves are also shown in fig .
[ fig : energy_fit ] ( a ) and ( b ) . for planar anchoring @xmath55 ; for homeotropic anchoring @xmath56 , i.  e. , in both cases the interaction potential is long - ranged . in case of homeotropic anchoring there is a critical size of colloidal particles ( of the order of interfacial width ) below which the discontinuous structural transition between the configurations with and without the defect disappears .
similar phenomena , named capillary bridging , can be observed in granular materials , liquid crystals , and binary mixtures  @xcite . in all cases ,
one of the phases ( otherwise metastable in the bulk ) condenses out and forms a bridge between the particles .
this transition is discontinuous and ends in a critical point for small particle radii . in our case , the nematic phase located between the isotropic phase and the defect forms a bridge between the particles and results in their attraction .
in conclusion , within the landau - de gennes formalism we have studied the effective pair interaction between two cylindrical parallel colloids trapped at the nematic - isotropic interface .
this pairwise interparticle force depends sensitively on the radii of colloids , the orientation of the director at the ni interface , and the anchoring strength , i.e. , the nematic elasticity is an essential ingredient of the interaction .
the formation or annihilation of defects influences the effective pair potential
. it also changes the repulsive character of the interaction at large separations to an attractive one at intermediate or small separations .
therefore , one should expect that many - body interactions are relevant to describe collective colloidal ordering at the interface .
another implication of our work is that the interface is able to rearrange the nucleation centers of , for example , phase - separating polymer network and help to design composite materials of extraordinary properties .
the landau - de gennes free energy is used to study theoretically the interaction of parallel cylindrical colloidal particles trapped at a nematic - isotropic interface . 
 we find that the effective interaction potential is non - monotonic . 
 the corresponding force - distance curves exhibit jumps and hysteresis upon approach / separation due to the creation / annihilation of topological defects . 
 minimization results suggest a simple empirical pair potential for the effective colloid - colloid interaction at the interface . 
 we propose that the interface - mediated interaction can play an important role in self - organization and clustering of colloidal particles at such interfaces .