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the presence of a surfactant film at a fluid - fluid interface alters the dynamics of the interface . this is manifested in behavior of the interfacial waves , induced either externally or by thermal fluctuations @xcite . the interfacial dynamics can be probed by measuring the light scattered on such surface waves ( see the review by earnshaw @xcite ) . the scattering of light on surface waves is a powerful tool for probing the properties of surfactant films at fluid interfaces @xcite , and a variety of systems have been recently investigated using this method ( e.g.  refs  @xcite , see also the review by cicuta and hopkinson @xcite ) . recently , the application of surfactant films to modify the interfacial properties has been extended to the systems in which one of the fluids is in liquid - crystalline phase ( e.g.  liquid crystal colloids @xcite ) . the presence of a liquid crystal as one of the fluids complicates the problem of probing the interfacial properties by studying the dynamics of the surface waves for the following reasons . firstly , there are additional degrees of freedom in the bulk of the liquid crystal phase due to its anisotropy . secondly , the interaction with the surfactant film is more complicated due to anisotropic anchoring . finally , the surfactant film in the anisotropic field created by the neighboring liquid crystal can itself show anisotropic behavior , even if it behaves as an two - dimensional isotropic fluid at the boundary between isotropic fluids .    a promising new direction for chemical and biological sensing devices has recently emerged which utilizes the properties of surfactant films self - assembled on the interface between water and a nematic liquid crystal . the surfactant film induces preferred orientation of the nematic director @xcite . the adsorption of chemical or biological molecules at such interface can then lead to reorientation of the nematic director , enabling detection by an imaging system @xcite .    in these methods , easy detection is limited to the systems in which adsorption changes anchoring properties of the interface with respect to the adjacent liquid crystal phase quite considerably . namely , the equilibrium anchoring angle should change in magnitude . the range of application of these systems could be made significantly broader , however , if a method were used that was sensitive to changes in the anchoring properties of the interface that did not necessarily result in nematic director reorientation . for example , the anchoring orientation may remain unchanged @xcite , the adsorption only changing the strength of the anchoring . if a small amount of an analyte is present in the water it may be adsorbed at the surfactant layer , provided the surfactant molecules possess appropriate chemical properties . generally , such adsorption will result in a change in the elastic and viscous properties of the interface . hence sensitive experiments which are able to determine the interfacial properties will allow much more detailed experimental insight into the properties of the interaction between the surfactants and the analyte than has hitherto been available , and experimental study of surface waves is a possible technique for this purpose . the theoretical description of surface waves at interfaces between nematic and isotropic liquids was made back in 1970s @xcite . the results demonstrated that the spectrum of surface waves has a more complicated structure than in the isotropic case , and allows the use surface scattering experiments to determine properties of nematic interfaces @xcite . since then , several theoretical and experimental advances have been made , and presently these systems remain a subject of investigation @xcite . the present paper presents a theoretical study of the dispersion of the surface waves at a monomolecular surfactant film between an isotropic liquid ( e.g.  water ) and a nematic liquid crystal.the main distinguishing features of such interfaces , are ( i ) the anchoring induced by the surfactant layer , ( ii ) the curvature energy of the interface , ( iii ) reduction of surface tension due to surfactant , and ( iv ) the anisotropy of the surface viscoelastic coefficients . we base our treatment on the mechanical model for anisotropic curved interfaces by rey @xcite , which takes into account anchoring and bending properties of the surfactant . we consider the case of the insoluble surfactant film that is in its most symmetric phase ( isotropic two - dimensional fluid ) , and induces homeotropic ( normal to the surface ) orientation of the director . the paper is organized as follows . the continuum model used in the rest of the paper is set up in section  [ sec : model ] . in section  [ sec : dispersion ] the dispersion relation for surface waves is derived . in section  [ sec : modes ] the numerical solution of the dispersion relation is solved with typical values of material parameters , and dispersion laws for different surface modes are analyzed in absence of the external magnetic field , and the influence of the magnetic field is discussed in section  [ sec : field ] . the explicit form of the dispersion relation is written in appendix  [ app : dispersion ] . in this section we formulate the model of the surfactant - laden interface between an isotropic liquid and a nematic liquid crystal , used in the present paper , and write down the governing equations . we base our treatment upon the models of the nematic - isotropic interface by rey @xcite , and well known hydrodynamic description of isotropic liquids @xcite and nematic liquid crystals @xcite . we consider the case when the surfactant film induces homeotropic ( normal to the surface ) orientation of the nematic director , which is usually true in a range of the surfactant concentrations @xcite . this case is the simplest to analyze , and , at the same time , the most important for biosensing applications where the direct change in anchoring angle can not be always observed . we include optional external magnetic field in our study and limit our analysis by considering the direction of the magnetic field that does not change equilibrium orientation of the nematic director . we assume that the system is far enough from any phase transitions both in the surfactant film @xcite and in the nematic phase @xcite . thus we avoid complications related to the fluctuations of the nematic and surfactant order parameters and the divergence of viscoelastic parameters near phase transitions . the surfactant films can exhibit rich phase behavior @xcite , and the form of the surface stress tensor depends upon the symmetry of the interface . however , this does not normally influence much the dispersion laws of the surface modes compared to the isotropic case @xcite . in the present paper we assume that the surfactant film is in the most symmetric phase ( isotropic two - dimensional fluid ) . although the symmetry of the film should break in presence of the adjacent liquid - crystalline bulk phase , the film remains isotropic in equilibrium if the anchoring of the nematic is homeotropic , and symmetry breaking can occur only due to fluctuations of the director field . if we introduce the order parameter for the film , the corresponding anisotropic contributions to the interfacial stress tensor would be of higher order in the fluctuations of the dynamic variables than is required in our linearized treatment , so such contributions can be omitted . we consider a surfactant layer at an interface between nematic and isotropic liquids to be macroscopically infinitely thin . we assume that the surfactant film is insoluble and newtonian . this means that the model is applicable to systems in which the interchange of surfactant molecules between the interface and adjacent bulk fluids is small , and the relaxation of the orientation of surfactant molecules is fast compared to relaxation of surface waves . we also assume heat diffusion to be sufficiently fast so that the system is in thermal equilibrium . we do not consider systems where other effects , such as polarity , are important . we shall choose coordinate system in such a way that the unperturbed interface lies at a plane @xmath0 , the half - space @xmath1 is occupied by the uniaxial nematic liquid crystal , and the half - space @xmath2 is filled by the isotropic liquid . other details of the geometry used in the present paper are summarized in appendix  [ app : geometry ] . the central equations in the present section are the conditions for the balance of forces ( eq .  ( [ eq : forcebalance ] ) and torques ( eq .  ( [ eq : torquebalance ] ) at the interface . the explicit form of these equations depends upon the chosen macroscopic model , and the rest of this section is devoted to formulation of the model used in the present paper . the interfacial force balance equation is the balance between the interfacial force and the bulk stress jump : @xmath3 here @xmath4 is the force per unit area exerted by the interfacial stress @xmath5 , @xmath6 is the force per unit area exerted by the isotropic fluid , @xmath7 is the force per unit area exerted by the nematic liquid crystal , the subscript @xmath8 indicates that the bulk stress fields in the isotropic liquid , @xmath9 , and in the nematic , @xmath10 , are evaluated at the interface , @xmath11 is the unit vector normal to the interface and directed into the isotropic liquid . the interfacial torque balance equation can be cast as @xmath12 where @xmath13 is the interfacial torque arising due to surface interactions , @xmath14 is the torque exerted upon the interface by the adjacent nematic liquid crystal .    the explicit model for surface and bulk stresses and torques that enter eqs  ( [ eq : forcebalance ] ) and ( eq .  ( [ eq : torquebalance ] ) is expanded in the remainder of this section .      in this and the following subsections we summarize the equations for the surface stress tensor @xmath5 and surface torque vector @xmath13 . we represent these quantities as a sum of corresponding non - dissipative ( elastic ) and dissipative ( viscous ) contributions : @xmath15 @xmath16 to describe the non - dissipative contributions in the surface stress tensor , @xmath17 , and surface torque vector , @xmath18 , we use the equilibrium model proposed by rey @xcite , which is summarized below . rey considered the interface with the helmholtz free energy per unit mass @xmath19 of the form @xmath20 where @xmath21 is the surface mass density , @xmath22 is the second fundamental tensor of the interface ( see appendix  [ app : geometry ] ) . the corresponding differential was written as @xmath23 where @xmath24_{\mathbf k,\mathbf b}\ ] ] is the interfacial tension , @xmath25 is the tangential component of the capillary vector ( @xmath26 is the surface projector ) , and @xmath27 is the bending moment tensor . the elastic surface stress tensor was found to be @xmath28 where the tangential surface molecular field is given by @xmath29 @xmath30 is surface gradient operator , @xmath31 denotes variational derivative with respect to @xmath11 . the elastic contribution to surface torque was written as @xmath32 where @xmath33 is the surface couple stress , @xmath34 is the levi - civita tensor , and @xmath35 is the surface alternator tensor . the viscous properties of interfaces between an isotropic fluid and a nematic liquid crystal were considered in detail by rey @xcite , and the results are summarized below . the forces and fluxes that contribute to the dissipation function @xmath36 were identified as follows : @xmath37 where @xmath38 and @xmath39 are , correspondingly , symmetric and antisymmetric parts of the surface viscous stress tensor @xmath40 , @xmath41 and @xmath42 are the components of the surface viscous molecular field tangential and normal to the surface , @xmath43\ ] ] is the surface rate - of - deformation tensor ( @xmath44 denotes the transposed tensor ) , @xmath45\ ] ] is the surface vorticity tensor , @xmath46 is surface velocity , @xmath47 and @xmath48 are the total time derivatives of the components @xmath49 and @xmath50 of the nematic director field @xmath51 , tangential and normal to the surface , correspondingly . generally , presence of the surfactant film at the interface complicates the form of the entropy production due to additional internal degrees of freedom of the surfactant , and to the anisotropy of the adjacent nematic liquid . however , if the surfactant film that is in its isotropic liquid phase and favors homeotropic anchoring of the nematic , the resulting anisitropic terms in the entropy production introduce corrections to the hydrodynamic equations of higher order than linear , and therefore can be neglected in the linearized treatment . since this is the case we are considering , we shall adopt the form of the entropy production ( [ eq : entropyproduction ] ) in our model and use the form of the viscous contribution to the surface stress tensor derived by rey @xcite , which is given by @xmath52,\end{aligned}\ ] ] where @xmath53 is the surface jaumann ( corrotational ) derivative @xcite of the tangential component of the director @xmath54 , and @xmath55 , @xmath56 are nine independent surface viscosity coefficients . in the isotropic case @xmath57 , the expression for the surface viscous stress tensor reduces to the viscous stress tensor of boussinesq - schriven surface fluid @xcite with the interfacial shear viscosity @xmath58 given by @xmath59 and dilatational viscosity @xmath60 given by @xmath61    the surface viscous torque , corresponding to eq . ( [ eq : entropyproduction ] ) , is given by @xcite @xmath62 where the surface viscous molecular field @xmath63 is @xmath64 the viscosity coefficients @xmath65 can be expressed in terms of quantities @xmath66 . we shall need only the expression for the tangential rotational viscosity : @xmath67      to calculate explicitly the interfacial tension @xmath68 ( eq . ( [ eq : def - tau ] ) ) , the tangential component of the capillary vector @xmath69 ( eq .  ( [ eq : def - xi ] ) ) , and the bending moment tensor @xmath70 ( eq .  ( [ eq : def - m ] ) ) , we need to know the dependence of the surface free energy @xmath19 on the orientation of the interface given by unit normal vector @xmath11 , and on its curvature described by second fundamental tensor @xmath22 . for small deviations of @xmath11 and @xmath22 from equilibrium , we can expand the free energy in powers of these quantities and truncate the series . the result can be represented as @xmath71 each of the contribution described below . the contribution @xmath72 corresponds to the surface tension @xmath73 of the equilibrium interface ( flat interface , adjacent nematic director normal to the interface ) : @xmath74    the anchoring contribution to the surface free energy density , @xmath75 , describes the energetics of the preferred alignment direction of the nematic director relative to the interface . for the homeotropic equilibrium anchoring , it can be written in terms of @xmath54 as follows : @xmath76 such expansion applied to the widely used rapini - papoular form of the anchoring free energy density @xcite @xmath77 shows that these definitions of the anchoring strength coefficient have opposite signs : @xmath78 we shall use @xmath79 as the anchoring strength coefficient to ensure that it is positive in the case of the homeotropic anchoring being considered . the third contribution to the surface free energy density , @xmath80 , is caused by finite interface thickness , and is related to the difference of the curvature of a surfactant film from the locally preferred ( spontaneous ) value . the widely used form of this contribution is the helfrich curvature expansion @xcite @xmath81 here the geometry of the interface is described by the mean curvature @xmath82 and the gaussian curvature @xmath83 , and the material parameters characterizing the interface are the bending rigidity @xmath84 , the saddle - splay ( or gaussian ) rigidity @xmath85 , and the spontaneous curvature @xmath86 . the term @xmath87 guarantees that the curvature energy of a flat interface ( @xmath88 , @xmath89 ) is zero .      to complete the description of the interface , we need the continuity equation for the surfactant concentration @xmath90 . for insoluble surfactants , the continuity equation reads : @xmath91    we shall extend the description of the dependence of the interfacial tension upon the concentration of surfactant , presented by buzza @xcite , to other parameters characterizing the interface ( surface tension @xmath73 , anchoring strength @xmath79 , bending rigidity @xmath84 , saddle - splay rigidity @xmath85 , spontaneous curvature @xmath86 , and surface viscosities @xmath66 , @xmath92 ) . for small deviation @xmath93 of the surfactant concentration @xmath90 from its equilibrium value @xmath94 , these coefficients can be written in form @xmath95 and similarly for other quantities . casting surface velocity @xmath46 as the time derivative of the small surface displacement @xmath96 , @xmath97 we obtain from the continuity equation eq . ( [ eq : continuitys ] ) that @xmath98 this allows us to represent the material parameters of the interface as @xmath99 @xmath100 @xmath101 @xmath102 in these formulas @xmath103 , @xmath104 , @xmath105 , @xmath106 are , correspondingly , the interfacial tension , anchoring strength , bending rigidity , and spontaneous curvature in the unperturbed interface , @xmath107 is the static dilatational elasticity , @xmath108 , @xmath109 , and @xmath110 are coefficients in the first order term of the expansion of anchoring strength , bending rigidity , and spontaneous curvature in powers of ( @xmath111 ) . there are similar expansions for gaussian rigidity @xmath85 and surface viscosities @xmath66 , @xmath92 . magnetic field @xmath112 in the isotropic and nematic regions satisfies maxwell equations @xcite @xmath113 @xmath114 neglecting magnetization of the interface , the boundary conditions read @xmath115 @xmath116 here the magnetization of the isotropic liquid is @xmath117 where @xmath118 is the magnetic permeability of the isotropic liquid , the magnetization of the uniaxial nematic liquid crystal is @xcite @xmath119 where @xmath120 is the difference of the longitudinal and transversal magnetic permeabilities of the nematic : @xmath121      we assume both the isotropic liquid and the nematic liquid crystal are incompressible , so that their densities @xmath122 and @xmath123 , are constant .    the linearized equations for the incompressible isotropic liquid are well known @xcite . they are the continuity equation @xmath124 and navier - stokes equations @xmath125 where the hydrodynamic stress tensor is given by @xmath126 where @xmath127 is the shear viscosity of the isotropic liquid , @xmath128 is the unit tensor , @xmath129\ ] ] is the strain rate tensor . we assume the non - slip boundary condition for the velocities of bulk fluids adjacent to the interface , which means the equality of the velocity of surfactant , @xmath46 , and that of the bulk fluids at an interface , @xmath130 : @xmath131      to describe the dynamics of the nematic liquid crystal that is far from the isotropic - nematic transition and has small deviations from its equilibrium state , we shall use the linearized form of the eriksen - leslie theory @xcite . the linearized equations for the incompressible nematic liquid crystal are the continuity equation ( [ eq : continuity ] ) , the equation for the velocity @xmath132 and the equation for the director @xmath133 here @xmath134\ ] ] is the antisymmetric vorticity tensor , @xmath135 is the reactive material parameter , @xmath136 is orientational viscosity , @xmath137 is the molecular field which , assuming frank form of the elastic free energy of a nematic liquid crystal in magnetic field @xcite @xmath138 ^ 2 \\ & + & \frac{k_3}2\left[\mathbf n\times\left(\nabla\times\mathbf n\right)\right]^2 -\frac12\chi_a(\mathbf n\cdot\boldsymbol{\mathcal h})^2,\end{aligned}\ ] ] has the linearized form @xmath139 where @xmath140\right\}+ \\ n_0\times\left[\mathbf n_0\times \left(\nabla\times\delta\mathbf n\right)\right]\right\ } + \chi_a(\mathbf n\cdot\boldsymbol{\mathcal h})\boldsymbol{\mathcal h},\end{aligned}\ ] ] @xmath141 , @xmath142 , and @xmath143 are the splay , twist , and bend frank elastic constants , correspondingly . the stress tensor can be represented as a sum of reactive and viscous ( dissipative ) contributions , @xmath144 the linearized form of the reactive part is @xmath145 the linearized viscous stress tensor of incompressible nematic is @xmath146 the quantities @xmath147 , @xmath148 , @xmath149 , @xmath136 , and @xmath135 can be expressed through more commonly used leslie viscosity coefficients @xcite . note that equating @xmath150 recovers the viscous stress tensor @xmath151 of the isotropic incompressible fluid ( last term in eq . ( [ eq : sigmai ] ) ) . the aim of this section is to construct the dispersion relation for the surface waves on the basis of the model set up above . we consider a surface wave with frequency @xmath152 and wavevector @xmath153 propagating along @xmath154 axis , and solve force balance equation , eq . ( [ eq : forcebalance ] ) , and torque balance equation , eq . ( [ eq : torquebalance ] ) using linearized form of the hydrodynamic equations written in section  [ sec : model ] .    in order to linearize the hydrodynamic equations , we represent pressure @xmath155 and the nematic director @xmath156 , where @xmath157 is the position in space , @xmath158 is time , in form @xmath159 @xmath160 where @xmath161 and @xmath162 are the deviations of pressure and director from their equilibrium values @xmath163 and @xmath164 , correspondingly . the velocity @xmath165 is itself the deviation from zero equilibrium velocity . homeotropic anchoring corresponds to @xmath166    for small deviations from the equilibrium , we shall use the hydrodynamic equations linearized in @xmath167 , @xmath161 , and @xmath162 . we shall assume these quantities to be independent of the coordinate @xmath168 ( @xmath169 ) and vanish at @xmath170 . the magnetic field can be also represented as @xmath171 , where @xmath172 is the equilibrium value , and the deviation @xmath173 can be found from the linearized form of the maxwell equations ( [ eq : rotmaxwell ] ) , ( [ eq : divmaxwell ] ) . the terms in the final equations , containing @xmath173 , are of higher order than linear , so we shall use only the equilibrium value , and skip the ` 0 ' subscript , so that @xmath174 .    substituting the interfacial free energy density ( [ eq : fs ] ) into eqs  ( [ eq : def - tau ] ) , ( [ eq : def - xi ] ) , ( [ eq : def - m ] ) , and ( [ eq : def - hse ] ) , we find the contributions up to the first order in @xmath96 ( and its derivatives ) and @xmath54 into surface tension @xmath175 bending moment tensor @xmath176\mathbf i_s -\bar\kappa\mathbf b,\ ] ] tangential component of the capillary vector @xmath177 and tangential surface molecular field @xmath178 + \nabla_s\cdot\left(\bar\kappa\mathbf b\right).\ ] ] the non - vanishing components of the surface viscous stress tensor ( [ eq : sigmasv ] ) are @xmath179 the total interfacial force @xmath180 can be found by substituting eqs  ( [ eq : sigmas ] ) , ( [ eq : sigmase ] ) , and ( [ eq : expl - tau])([eq : expl - sigmasv ] ) into eq . ( [ eq : def - forces ] ) , and has components @xmath181 where @xmath182 to write the explicit form of the force balance equations ( [ eq : forcebalance ] ) , we also need the expressions for the components of the force ( [ eq : def - forcei ] ) exerted by the isotropic fluid , @xmath183 and the components of the force ( [ eq : def - forces ] ) exerted by the nematic liquid crystal , @xmath184_{z=-0 } , \\ f^n_y&=&\left[\frac{1+\lambda}2h_y-\nu_3\partial_zv_y\right]_{z=-0 } , \\ f^n_z&=&\left(p-2\nu_1\partial_zv_z\right)_{z=-0},\end{aligned}\ ] ] the hydrodynamic fields @xmath167 , @xmath185 , @xmath51 in the bulk isotropic and nematic liquids are found by solution of the hydrodynamic expressions . the explicit formulas are presented in appendices  [ app : isotropic ] and [ app : nematic ] . next we introduce fourier transforms in the @xmath154 coordinate and in time as @xmath186 @xmath187 @xmath188 ( for brevity we shall henceforth omit arguments of the transformed functions ) . performing fourier - transform of the force balance equation ( [ eq : forcebalance ] ) , and substituting @xmath189 , we obtain balance equations for the force components in form @xmath190c_i^{n\vert}+ \\ \nonumber + i\omega\frac{1+\lambda}2 \sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right ] b_i^\vert c_i^{n\vert } \\ \label{eq : balancevx } = 0,\quad\end{aligned}\ ] ] @xmath191 b_i^\bot c_i^{n\bot } \\ \label{eq : balancevy } = 0,\end{aligned}\ ] ] @xmath192 where @xmath193 is the complex dilatational modulus , @xmath194 is defined in appendix  [ app : isotropic ] by eq . ( [ eq : mi ] ) , and the quantities @xmath195 , @xmath196 , @xmath197 , @xmath198 , @xmath199 , @xmath200 , and @xmath201 are defined in appendix  [ app : nematic ] by eqs  ( [ eq : mvert ] ) , ( [ eq : mbot ] ) , ( [ eq : cvert ] ) , ( [ eq : cbot ] ) , ( [ eq : bvert ] ) , ( [ eq : bbot ] ) , and ( [ eq : a ] ) , correspondingly .    to write the interfacial torque balance equation ( [ eq : torquebalance ] ) , we cast the torque exerted upon the interface by the nematic liquid crystal , @xmath14 , and the interfacial torque arising due to surface interactions , @xmath13 , entering the interfacial torque balance equation ( [ eq : torquebalance ] ) , in form @xmath202 and @xmath203 where the molecular field from the bulk @xmath204_s\ ] ] has linearized components @xmath205 @xmath206 and the surface molecular field @xmath207 can be represented as a sum of elastic ( @xmath208 ) and viscous ( @xmath63 ) contributions @xmath209 given by eqs  ( [ eq : def - hse ] ) and ( [ eq : def - hsv ] ) , correspondingly , and can be represented in components as @xmath210 @xmath211 @xmath212 @xmath213 then the surface torque balance equations can be written as @xmath214 @xmath215 or , substituting the expressions ( [ eq : solvz ] ) , ( [ eq : solnx ] ) and ( [ eq : solny ] ) , @xmath216c^{n\vert}_i&=&0,\end{aligned}\ ] ] @xmath217    the interfacial force balance equations ( eqs  ( [ eq : balancevx])([eq : balancevz ] ) ) and the interfacial torque balance equations ( eqs  ( [ eq : balancenx])([eq : balanceny ] ) ) form , with account of eqs  ( [ eq : cbot ] ) and ( [ eq : cvert ] ) , a homogeneous system of linear algebraic equations in @xmath218 , @xmath219 , @xmath220 , @xmath221 and @xmath222 . the dispersion relation is obtained from the condition of existence of a solution to these equations , i.e. the requirement for the determinant @xmath223 of the matrix of coefficients for this system to be zero @xmath224    the equations ( [ eq : balancevy ] ) and ( [ eq : balanceny ] ) in @xmath219 and @xmath222 decouple from the equations ( [ eq : balancevx ] ) , ( [ eq : balancevz ] ) and ( [ eq : balancenx ] ) in @xmath218 , @xmath220 and @xmath225 . therefore , the matrix of coefficients is block - diagonal , and the dispersion relation ( eq . ( [ eq : d ] ) ) is equivalent to a pair of relations for @xmath226 and @xmath168 directions : @xmath227 @xmath228 where @xmath229 is the determinant of the @xmath230 matrix @xmath231 of coefficients for the equations ( [ eq : balancevx ] ) , ( [ eq : balancevz ] ) and ( [ eq : balancenx ] ) , and @xmath232 is the determinant of the @xmath233 matrix of coefficients for the equations ( [ eq : balancevy ] ) and ( [ eq : balanceny ] ) .    the explicit form of the dispersion relations is presented in appendix  [ app : dispersion ] and can be readily used for the numerical analysis of surface modes . in this section the dispersion equation , which is presented in appendix  [ app : dispersion ] , is solved numerically , and surface modes of different types are analyzed . for simplicity , we assume the density of the isotropic liquid , @xmath122 , to be small enough to be neglected ( e.g.  nematic  surfactant  air interface ) . we also assume that the magnetic field is absent . the surface modes can be easily classified at low wavevectors @xmath234 . expansion of the dispersion relation in powers of the wavevector @xmath234 is a straightforward exercise in algebra , and the resulting modes are described below . firstly , there is a transverse capillary mode , which has the dispersion law similar to that in the case of an isotropic liquid - liquid interface @xcite : @xmath235 the principal contribution to this mode at large wavelengths arises due to the restoring influence of surface tension @xmath236 , and the predominant motion is in the direction normal to the interface ( @xmath237 ) . the differences from the isotropic case , related to anisotropy of viscous dissipation in the nematic , appear in higher orders in @xmath234 . the dilatational ( or compressional ) mode with predominant motion in the direction along wave propagation ( @xmath154 ) arises in presence of surfactant layer due to the restoring force provided by the dilatational elastic modulus @xmath238 . the dispersion law for this mode can be written as @xmath239^{1/3 } + o\left(q^{4/3}\right),\ ] ] where the miesowicz viscosity @xmath240 is given by @xcite @xmath241 the difference from the dispersion law for the dilatational mode in the case of a surfactant film at the interface between isotropic fluids , given by @xcite @xmath242 arises due to anisotropy of viscous dissipation in nematic . a new mode , specific to the nematic , is driven by relaxation of the director field to equilibrium due to anchoring at the interface and has the disperion law @xmath243 such relaxation is present even in absence of motion of the interface ( e.g.  when the interface is solid ) , so that @xmath244 does not vanish at @xmath245 . for nematic - isotropic interfaces , the corresponding motion of the interface is induced by backflow effects . finally , behavior of the in - plane shear mode , with motion in @xmath168-direction , is also governed by relaxation of the nematic director due to anchoring . the corresponding dispersion law @xmath246 appears to be different from the isotropic case , where the damping of the in - plane shear mode in absence of anchoring is governed by the surface viscosity @xmath58 @xcite . gravity @xmath247 , so far neglected in our analysis , becomes important at wavevectors @xmath248 and can be taken into account by adding the hydrostatic pressure term @xmath249 to eq . ( [ eq : m22 ] ) , which corresponds to the additional contribution @xmath250 to the vertical component of the force , eq . ( [ eq : fsz ] ) . the resulting dispersion law for transversal mode is given by expression @xmath251 which describes well - known gravity waves @xcite .    in the opposite case of large wavevectors , the curvature energy becomes important . analysis of eqs  ( [ eq : fsz ] ) and ( [ eq : tsx ] ) yields the characteristic values of @xmath234 @xmath252 and @xmath253 below which one can neglect in the dispersion relation the terms containing bending rigidity @xmath84 and its derivatives with respect to surfactant concentration , given by @xmath254 . usually @xmath255 , and the range of @xmath234 in which both gravity and curvature contributions become small , given by @xmath256 is rather wide . for typical values @xmath257kg / m@xmath258 , @xmath259m / s@xmath260 , @xmath261j / m@xmath260 , @xmath262j , the equation ( [ eq : good - q ] ) reads @xmath263@xmath264@xmath265 , which includes the range of wavevectors typically probed by surface light scattering experiments .    to obtain the dispersion laws for surface modes at larger values of the wavevector @xmath234 , the dispersion equation must be solved numerically . the numerical solution presented below uses the following typical values of the material parameters when it is not indicated otherwise . for the nematic liquid crystal we use the parameters of 4-@xmath266-pentyl-4-cyanobiphenyl ( 5cb ) at 26@xmath267c @xcite : the density @xmath268kg / m@xmath269 , the elastic constants @xmath270n , @xmath271n , @xmath272n , the leslie viscosities @xmath273kg/(m@xmath274s ) , @xmath275kg/(m@xmath274s ) , @xmath276kg/(m@xmath274s ) , @xmath277kg/(m@xmath274s ) , @xmath278kg/(m@xmath274s ) , @xmath279kg/(m@xmath274s ) . the viscosity coefficient used in the present paper can be calculated from the leslie equations @xcite and equal @xmath280kg/(m@xmath274s ) , @xmath281kg/(m@xmath274s ) , @xmath282kg/(m@xmath274s ) , @xmath283kg/(m@xmath274s ) , @xmath284 . we use the value of the bending rigidity @xmath285j which is typical for surfactant layers @xcite . for other parameters we use the following typical values : @xmath286kg / s , @xmath287 , @xmath288n / m , @xmath289n / m , @xmath290j / m@xmath291 , @xmath292j / m@xmath291 . dispersion law @xmath293 for different surface modes in absence of gravity , obtained by solution of the dispersion relation ( [ eq : dvert ] ) with the values of the parameters given in the text . numbers 1 , 2 , 3 denote transverse , dilatational , and nematic director relaxation modes , correspondingly . prime and double prime denote real ( solid line ) and imaginary ( dashed line ) parts of @xmath152 , correspondingly . ]    the dispersion law @xmath293 for different surface modes in absence of gravity , obtained by solution of the dispersion relation ( [ eq : dvert ] ) with the values of the parameters given above , is presented in figure  [ fig : nogravity ] . at low @xmath234 the dispersion of for modes 1 , 2 , 3 , as denoted figure  [ fig : nogravity ] , is in good agreement with approximate formulas ( [ eq : omegac ] ) , ( [ eq : omegad ] ) , and ( [ eq : omegan ] ) , correspondingly . the noticeable discrepancy in behavior of capillary and dilatational modes appears at @xmath294@xmath265 , and the damping of surface waves becomes large at larger @xmath234 , which is qualitatively similar to the case of the interface between isotropic liquids . the results presented in figure  [ fig : nogravity ] suggest that in the typical range of @xmath234 probed by surface light scattering experiments ( @xmath295@xmath296@xmath265 ) , the approximate expressions ( [ eq : omegac ] ) , ( [ eq : omegad ] ) do not describe well the dispersion curves , and accurate solution of the dispersion equation should be used instead . dispersion law @xmath293 for different surface modes in presence of gravity @xmath297m / s@xmath260 , obtained by solution of the dispersion relation ( [ eq : dvert ] ) with the values of the parameters given in the text . numbers 1 , 2 , 3 denote transverse , dilatational , and nematic director relaxation modes , correspondingly . prime and double prime denote real ( solid line ) and imaginary ( dashed line ) parts of @xmath152 , correspondingly . vertical dotted line corresponds to the value of @xmath298 given by eq . ( [ eq : qg ] ) . ] figure  [ fig : gravity ] presents the dispersion law @xmath293 for different surface modes obtained by solution of the dispersion relation ( [ eq : dvert ] ) in presence of gravity @xmath297m / s@xmath260 . in agreement with the discussion above , the influence of gravity on the dispersion laws is small at @xmath299 , where @xmath298 is given by eq . ( [ eq : qg ] ) . dependence of the real ( solid line ) and imaginary ( dashed line ) parts of the frequency of the mode 1 ( as defined on figure  [ fig : nogravity ] ) upon the bending rigidity @xmath300 , calculated at @xmath301@xmath265 in absence of gravity . vertical line corresponds to the value of @xmath300 that satisfies eq . ( [ eq : qkappa ] ) . ] if the bending rigidity @xmath84 is large , its influence becomes noticeable , as it is demonstrated in figure  [ fig : bending ] . for @xmath302 , typical for surfactant films , the value of @xmath303 , given by eq . ( [ eq : qkappa ] ) , corresponds to wavelength close to atomic scales , and curvature energy can be neglected in typical surface light scattering experiments , in agreement with the discussion above . dependence of the real ( solid line ) and imaginary ( dashed line ) parts of the frequency @xmath152 of the mode 3 ( as defined on figure  [ fig : nogravity ] ) , normalized by @xmath304 ( see eq . ( [ eq : omegan ] ) ) , upon the anchoring strength @xmath305 , calculated at @xmath306@xmath265 in absence of gravity . ]     dependence of the real ( solid line ) and imaginary ( dashed line ) parts of the frequency @xmath152 of the in - plane shear mode , normalized by @xmath304 ( see eq . ( [ eq : omegas ] ) ) , upon the anchoring strength @xmath305 , calculated at @xmath306@xmath265 in absence of gravity . ]    the dispersion law for the modes governed by relaxation of the nematic director field in @xmath154 and @xmath168 directions due to anchoring of the nematic director at the interface , obtained by numerical solution of the dispersion equation with the values of the parameters given above , are well described by the equations ( [ eq : omegan ] ) and ( [ eq : omegas ] ) . however , as the anchoring strength becomes smaller , other mechanisms start to take over , as demonstrated in figures  [ fig : xanchoring ] and [ fig : yanchoring ] . in this section we discuss how the surface modes described in section  [ sec : modes ] are altered in presence of the external magnetic field directed normally to the surface ( along @xmath237 axis ) . the external magnetic field effectively acts on the nematic molecules as an additional molecular field ( see eq . ( [ eq : h - star ] ) ) , and the primary counteracting mechanism is provided by orientational shear relaxation . thus we may expect the influence of the magnetic field become noticeable at @xmath307     dependence of the real ( solid lines ) and imaginary ( dashed lines ) parts of the frequencies of the modes 1 and 2 ( as defined on figure  [ fig : nogravity ] ) upon the magnetic field , calculated at @xmath308@xmath265 in absence of gravity . vertical dotted line corresponds to @xmath309 ( see eq . ( [ eq : hstar ] ) ) . ] the results of the numerical solution of the dispersion equation in presence of magnetic field , presented in figure  [ fig : field ] , confirm that noticeable change in dispersion of capillary and dilatational modes arises only around the value of the field given by eq . ( [ eq : hstar ] ) . the change due to magnetic field in modes governed by anchoring is found to be negligibly small .    at low @xmath234 the dispersion of a capillary mode in strong magnetic field is different from the law ( [ eq : omegac ] ) and is given by @xmath310 the frequency of this mode becomes sensitive to the anchoring properties of the interface , because the nematic director tends to be oriented along the field rather than to be advected with the nematic liquid . the practical use of this effect is , however , limited , because at short wavelengths extremely large magnetic field is required , and at long wavelengths gravity becomes dominating ( eq . ( [ eq : omegag ] ) .    in principle , magnetic field can also influence surface waves through change in the properties of the interface ( e.g.  surface tension ) due to the magnetization of the surfactant . separate study is required to estimate the magnitude of this effect . we have obtained the dispersion relation for the surface waves at a surfactant - laden nematic isotropic interface for the case when the surfactant film induces homeotropic ( normal to the surface ) orientation of the director , and the surfactant film is in the isotropic two - dimensional fluid phase . we have analyzed the dispersion law of different surface modes analytically in long wavelength limit , and numerically in broader range of wave vectors , using typical values of the material parameters .    at long wavelengths the dispersion of capillary , dilatational ( or compression ) , in - plane shear , and director relaxation modes is described by equations ( [ eq : omegac ] ) ( or ( [ eq : omegag ] ) ) , ( [ eq : omegad ] ) , ( [ eq : omegas ] ) , and ( [ eq : omegan ] ) , correspondingly . at smaller wavelength , the solution of the full dispersion relation should be used . gravity influences the transversal mode at small wavevectors ( eq . ( [ eq : qg ] ) ) , and curvature energy of surfactant can be neglected if wavevector is not too large ( eq . ( [ eq : qkappa ] ) ) . for all modes , the influence of the external magnetic field directed normally to the interface is small . the influence of the magnetic field should be more pronounced if the direction of the field does not coincide with equilibrium nematic director . in this case the dispersion law for surface modes may be expected to be quantitatively different due to anisotropy of viscous dissipation in nematic , and different anchoring energy . the results of the present paper can be readily extended to the case of arbitrary direction of the external field and to other types of nematic anchoring . other possible developments , which may increase the range of accessible systems and conditions , is the extension of the results to wider range of the states of the surfactant film , and the study of the effects which may be caused by the phase transitions in the surfactant film and bulk liquid crystal . dependence of the dispersion waves upon the parameters of the interface suggests the surface light scattering on a surfactant - laden nematic - isotropic interface as a potential method for determining of the properties of surfactant - laden nematic - isotropic interfaces , and as a possible candidate for a chemical or biological sensing technique . i thank prof .  c.  m.  care for fruitful discussion of the results , and prof . p.  d.  i.  fletcher for the discussion about surfactant - laden nematic - isotropic interfaces which instigated this work . the geometrical description we use is similar to that of that presented in works @xcite and @xcite . we choose the plane @xmath0 to coincide with the unperturbed interface , the half - space @xmath1 to be occupied by the uniaxial nematic liquid crystal , and the half - space @xmath2 to be filled by the isotropic liquid . let the position of a fluid particle at the interface be @xmath311 , where @xmath312 is its position on the undeformed interface ( @xmath0 ) , and @xmath313 is the displacement vector with components @xmath314 . we shall use @xmath315 and @xmath316 as surface coordinates and denote them as @xmath317 , @xmath318 and other greek indices taking values 1 and 2 . the position @xmath319 of fluid particles at the interface in 3d space can be cast as @xmath320 the surface tangent base vectors @xmath321 , corresponding to the chosen surface coordinates , can be written in terms of the components of the displacement vectors : @xmath322 and @xmath323 the surface metric tensor @xmath324 has determinant @xmath325 the corresponding reciprocal base vectors @xmath326 and metric tensor @xmath327 take form @xmath328 @xmath329 @xmath330 the base and reciprocal base vectors satisfy @xmath331    we write the unit vector @xmath11 , normal to the interface and directed into the isotropic liquid , as @xmath332 we shall also define the dyadic surface idem factor @xmath333 the surface gradient operator @xmath334 and the second fundamental tensor @xmath335 the mean curvature @xmath82 and gaussian curvature @xmath83 are given by @xmath336 @xmath337    other useful identities include the surface projection @xmath54 of a nematic director field @xmath51 , eqs  ( [ eq : n ] ) and ( [ eq : n0 ] ) , @xmath338 and its surface divergence @xmath339 this appendix presents the solution to the linearized hydrodynamic equations in bulk isotropic liquid , obtained by kramer @xcite . substitution of eq . ( [ eq : vfourier ] ) into eq . ( [ eq : continuity ] ) yields @xmath340 substituting eqs  ( [ eq : vfourier ] ) and ( [ eq : pfourier ] ) into eqs  ( [ eq : navier - stokes])([eq : s ] ) , we obtain @xmath341 \tilde v_x&=&iq\tilde p , \\ \label{eq : kramer - vy } \left[i\omega\rho^i+\eta\left(q^2-\partial_z^2\right)\right ] \tilde v_y&=&0 , \\ \label{eq : kramer - vz } \left[i\omega\rho^i+\eta\left(q^2-\partial_z^2\right)\right ] \tilde v_z&=&-\partial_z\tilde p,\end{aligned}\ ] ] where equation ( [ eq : kramer - vy ] ) is decoupled from other equations . the general solution to eqs  ( [ eq : continuity - fourier])([eq : kramer - vz ] ) vanishing at @xmath342 can be written as @xmath343 @xmath344 @xmath345 @xmath346 with @xmath347 the quantities @xmath348 , @xmath349 , and @xmath350 are functions of @xmath234 and @xmath152 and are determined by the boundary conditions at the interface as follows : @xmath351 @xmath352 @xmath353 where the superscript @xmath354 indicates that the values of the corresponding dynamic variables are taken at @xmath355 . in this appendix the solution is presented to the linearized hydrodynamic equations in bulk nematic liquid crystal .    for the equilibrium director along @xmath237 axis ( eq .  ( [ eq : n0 ] ) ) the fourier - transform , similar to equations ( [ eq : vfourier])([eq : nfourier ] ) , of the linearized molecular field ( eq .  ( [ eq : h ] ) ) , @xmath356 , has non - zero components @xmath357 @xmath358 substituting them into eqs  ( [ eq : eriksenleslie ] ) , ( [ eq : sigman])([eq : sigmanv ] ) , we obtain the following linear differential equations , @xmath359\tilde v_x-\frac{1-\lambda}2\partial_z\tilde h_x = iq\tilde p,\ ] ] @xmath360\tilde v_y -\frac{1-\lambda}2\partial_z\tilde h_y=0,\ ] ] @xmath361 \tilde v_z - iq\frac{1+\lambda}2\tilde h_x=-\partial_z\tilde p,\ ] ] which are analogous to eqs  ( [ eq : kramer - vx])([eq : kramer - vz ] ) for isotropic liquids . equation ( [ eq : dndt ] ) for the director after fourier transform gives two equations , @xmath362 and @xmath363 where @xmath364 and @xmath365 are given by eqs  ( [ eq : hx ] ) and ( [ eq : hy ] ) . thus we have six linear differential equations ( eqs  ( [ eq : continuity - fourier ] ) and ( [ eq : kramer - vxn])([eq : kramer - nyn ] ) ) for six dynamic variables ( pressure , three components of velocity , and two components of director ) . equations ( [ eq : kramer - vyn ] ) and ( [ eq : kramer - nyn ] ) for @xmath366 and @xmath367 decouple from the others , their general solution vanishing at @xmath368 can be cast as @xmath369 @xmath370 where @xmath371},\ ] ] @xmath372 and @xmath373 , @xmath374 , are the roots of the quadratic equation @xmath375 where @xmath376 @xmath377 @xmath378    the general solution to the equations ( [ eq : continuity - fourier ] ) , ( [ eq : kramer - vxn ] ) , ( [ eq : kramer - vzn ] ) , and ( [ eq : kramer - nxn ] ) vanishing at @xmath368 can be cast as @xmath379 @xmath380 @xmath381 @xmath382 where @xmath383 b^\vert_i+ \\ + \left . \left[i\omega\rho^n+\nu_3q^2 -\left(2\nu_1-\nu_3\right)\left(m^{n\vert}_i\right)^2\right ] \right\},\end{aligned}\ ] ] @xmath384 @xmath385 and @xmath386 , @xmath387 , are the roots of the cubic equation @xmath388 where @xmath389 @xmath390 q^2\right\}k_3,\end{aligned}\ ] ] @xmath391 \\ \nonumber -\left\{i\omega\rho^n -\left[\frac{1+\lambda^2}2\gamma_1 - 2\left(\nu_1+\nu_2-\nu_3\right)\right]q^2 \right\ } \\ \nonumber \times\left(k_1q^2-\chi_a\mathcal h^2\right ) \\ -\left[i\omega\rho^n -\left(\frac{1-\lambda^2}4\gamma_1-\nu_3\right)q^2 \right]k_3q^2,\end{aligned}\ ] ] @xmath392\left(k_1q^2-\chi_a\mathcal h^2\right)q^2.\end{aligned}\ ] ]    the quantities @xmath197 and @xmath198 are functions of @xmath234 and @xmath152 and are determined by the boundary conditions at the interface as @xmath393 @xmath394/\delta,\end{aligned}\ ] ] where @xmath395 and expressions for @xmath396 , @xmath397 , and @xmath398 are obtained from eqs  ( [ eq : cbot ] ) and ( [ eq : cvert ] ) by cyclic permutation of subscript indices . to write the explicit form of the dispersion relations ( [ eq : dvert ] ) and ( [ eq : dbot ] ) , we recast equations ( [ eq : cbot ] ) and ( [ eq : cvert ] ) in form @xmath399 @xmath400 where @xmath401 @xmath402 is given by eq . ( [ eq : delta ] ) , @xmath403 and @xmath404 are given by eqs  ( [ eq : bbot ] ) and ( [ eq : bvert ] ) , @xmath196 and @xmath195 are given by eqs  ( [ eq : mvert ] ) and ( [ eq : mbot ] ) , correspondingly . then the dispersion relation ( [ eq : dbot ] ) can be written as @xmath405 where @xmath406 is @xmath407 matrix of coefficients for equations ( [ eq : balancevy ] ) and ( [ eq : balanceny ] ) @xmath408 with the following components : @xmath409b^\bot_i\right\}l^{\left(vy\right)}_i,\end{aligned}\ ] ] @xmath410b^\bot_i\right\}l^{\left(ny\right)}_i,\end{aligned}\ ] ] @xmath411 @xmath412    the dispersion relation ( [ eq : dvert ] ) can be written as @xmath413 where @xmath414 is @xmath415 matrix of coefficients for equations ( [ eq : balancevx ] ) , ( [ eq : balancevz ] ) , and ( [ eq : balancenx ] ) @xmath416 with the following components : @xmath417 l^{\left(vx\right)}_i + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(vx\right)}_i,\end{aligned}\ ] ] @xmath418 l^{\left(vz\right)}_i + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(vz\right)}_i,\end{aligned}\ ] ] @xmath419 l^{\left(nx\right)}_i+ \\ \nonumber { } + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(nx\right)}_i,\end{aligned}\ ] ] @xmath420 @xmath421 @xmath422 @xmath423\right\}l^{\left(vx\right)}_i,\end{aligned}\ ] ] @xmath424\right\}l^{\left(vz\right)}_i,\end{aligned}\ ] ] @xmath425\right\}l^{\left(nx\right)}_i.\end{aligned}\ ] ] note that gravity @xmath247 has been incorporated into the dispersion relation by adding the hydrostatic pressure term @xmath249 to @xmath426 ( eq .  ( [ eq : m22 ] ) ) .    by setting @xmath427 , and setting to zero quantities @xmath141 , @xmath142 , @xmath143 , @xmath135 , @xmath136 , @xmath428 , and @xmath120 , specific to nematic , and neglecting curvature contributions by setting to zero @xmath300 and @xmath254 , the dispersion relation is reduced to the well studied form for the case of isotropic liquids @xcite .
a theoretical study is presented of surface waves at a monomolecular surfactant film between an isotropic liquid and a nematic liquid crystal for the case when the surfactant film is in the isotropic two - dimensional fluid phase and induces homeotropic ( normal to the interface ) orientation of the nematic director . the dispersion relation for the surface waves 
 is obtained , and different surface modes are analyzed with account being taken of the anchoring induced by the surfactant layer , the curvature energy of the interface , and the anisotropy of the viscoelastic coefficients . 
 the dispersion laws for capillary and dilatational surface modes retain structure similar to that in isotropic systems , but involve anisotropic viscosity coefficients . 
 additional modes are related to relaxation of the nematic director field due to anchoring at the interface . 
 the results can be used to determine different properties of nematic - surfactant - isotropic interfaces from experimental data on surface light scattering .